1 Introduction

We consider the defocusing nonlinear Schrödinger equation

$$\begin{aligned} (\text {NLS}) \ \ \left| \begin{array}{ll}i\partial _tu+\Delta u-u|u|^{p-1}=0,\\ u_{|t=0}=u_0, \end{array}\right. \ \ (t,x)\in [0,T_*)\times {\mathbb {R}}^d, \ \ u(t,x)\in {\mathbb {C}}.\nonumber \\ \end{aligned}$$
(1.1)

in dimension \(d\ge 3\) for an integer nonlinearity \(p\in 2{\mathbb {N}}^*+1\) and address the problem of its global dynamics. We begin by giving a quick introduction to the problem and its development.

1.1 Cauchy theory and scaling

It is a very classical statement that smooth well localized initial data \(u_0\) yield local in time, unique, smooth, strong solutions. For the global dynamics, two quantities conserved along the flow (1.1) are of the utmost importance:

$$\begin{aligned}&\qquad {\textit{mass:}}\qquad M(u)=\displaystyle \int _{{\mathbb {R}}^d}|u(t,x)|^2=\int _{{\mathbb {R}}^d} |u_0(x)|^2, \nonumber \\ {\textit{energy:}}&\quad E(u)=\displaystyle \frac{1}{2}\int _{{\mathbb {R}}^d}|\nabla u(t,x)|^{2}+\frac{1}{p+1}\int _{{\mathbb {R}}^d}|u(t,x)|^{p+1}dx=E(u_0).\nonumber \\ \end{aligned}$$
(1.2)

The scaling symmetry group

$$\begin{aligned} u_\lambda (t,x)=\lambda ^{\frac{2}{p-1}}u(\lambda ^2t,\lambda x), \ \ \lambda >0 \end{aligned}$$

acts on the space of solutions by leaving the critical norm invariant

$$\begin{aligned} \int _{{\mathbb {R}}^d}|\nabla ^{s_c}u_\lambda (t,x)|^2=\int _{{\mathbb {R}}^d}|\nabla ^{s_c}u(t,x)|^2\ \ \text{ for }\ \ s_c=\frac{d}{2}-\frac{2}{p-1}. \end{aligned}$$

Accordingly, the problem (1.1) can be classified as energy subcritical, critical or supercritical depending on whether the critical Sobolev exponent \(s_c\) lies below, equal or above the energy exponent \(s=1\). This classification also reflects the (in)/ability for the kinetic term in (1.2) to control the potential one via the Sobolev embedding \(H^1\hookrightarrow L^q\).

1.2 Classification of the dynamics

We review the main known dynamical results which rely on the scaling classification.

Energy subcritical case. In the energy subcritical case \(s_c<1\), the pioneering work of Ginibre–Velo [22] showed that for all \(u_0\in H^1\), there exists a unique strong solution \(u\in {\mathcal {C}}^0([0,T_*),H^1)\) to (1.1) and identified the blow up criterion

$$\begin{aligned} T_*<+\infty \Longrightarrow \mathop {\mathrm{lim}}_{t\uparrow T_*}\Vert u(t)\Vert _{H^1}=+\infty . \end{aligned}$$
(1.3)

Conservation of energy, which is positive definite and thus controls the energy norm \(H^1\), then immediately implies that the solution is global, \(T_*=+\infty \). In fact, it can be shown in addition that these solutions scatter as \(t\rightarrow \pm \infty \), [23].

Energy critical problem. In the energy critical case \(s_c=1\), the criterion (1.3) fails and the energy density could concentrate. For the data with a small critical norm, Strichartz estimates allow one to rule out such a scenario, [10]. The large data critical problem has been an arena of an intensive and remarkable work in the last 20 years.

For large spherically symmetric data in dimensions \(d=3,4\), the energy concentration mechanism was ruled out by Bourgain [7] and Grillakis [25] via a localized Morawetz estimate. In Bourgain’s work, a new induction on energy argument led to the statements of both the global existence and scattering. These results were extended to higher dimensions by Tao [60].

The interaction Morawetz estimate, introduced in [11], led to a breakthrough on the global existence and scattering for general solutions without symmetry, first in \(d=3\), [11], then in \(d=4\), [55], and \(d\ge 5\), [65].

A new approach was introduced in Kenig–Merle [30] in which, if there exists one global non-scattering solution, then using the concentration compactness profile decomposition [2, 46], one extracts a minimal blow up solution and proves that up to renormalization, such a minimal element must behave like a soliton. The existence of such objects is ruled out using the defocusing nature of the nonlinearity, which is directly related to the non existence of solitons for defocusing models.

In all of these large data arguments, the a priori bound on the critical norm provided by the conservation of energy plays a fundamental role. Let us note that in the energy critical focusing setting, the concentration of the critical norm is known to be possible via type II (non self similar) blow up with soliton profile, see e.g. [34, 41, 51,52,53,54].

Energy supercritical problem. In the energy supercritical range \(s_c>1\), local in time unique strong solutions can be constructed in the critical Sobolev space \(H^{s_c}\), [10, 32]. Kenig–Merle’s approach, [31, 32], gives a blow up criterion

$$\begin{aligned} T_*<+\infty \Longrightarrow \limsup _{t\uparrow T_*}\Vert u(t,\cdot )\Vert _{H^{s_c}}=+\infty , \end{aligned}$$

but the question of whether this actually happens for any solution remained completely open. On the other hand, the main difficulty in proving that \(T_*=\infty \) for all solutions is that there are no a priori bounds at the scaling level of regularity \(H^{s_c}\).

1.3 Qualitative behavior for supercritical models

The question of global existence or blow up for energy supercritical models is a fundamental open problem in many nonlinear settings, both focusing and defocusing. For focusing problems, the existence of finite energy type I (self similar) blow up solutions is known in various instances, see e.g. [15, 19, 36, 38], and solitons have been proved to be admissible blow up profiles in certain type II (non self-similar) blow up regimes in all three settings of heat, wave and Schrödinger equations, see e.g. [14, 28, 39, 42, 49]. There are also several examples of supercritical problems with positive definite energy (wave maps, Yang–Mills) which admit smooth self-similar profiles and thus provide explicit blow up solutions, [5, 18, 58].

On the other hand, for defocusing problems, soliton-like solutions are known not to exist and admissible self similar solutions are expected not to exist. For a simple defocusing model like the scalar nonlinear defocusing heat equation \(\partial _tu=\Delta u-|u|^{p-1}u,\) a direct application of the maximum principle ensures that \(L^\infty \cap H^1\) data yield uniformly bounded solutions which are global in time and in fact dissipate. We recall again that for the energy critical problems, blow up occurs in the focusing case, where solitons exist, and it does not in the defocusing case where solitons are known not to exist.

This collection of facts led to the belief, as explicitly conjectured by Bourgain in [6], that global existence and scattering should hold for the energy supercritical defocusing Schrödinger and wave equations. Indications of various qualitative behaviors supporting different conclusions have been provided (we give a highly incomplete list) in numerical simulations e.g. [12, 50], in model problems showing blow up e.g. [61, 62], in examples of global solutions e.g. [4, 33], in logarithmically supercritical problems e.g. [13, 59, 63], and in ill-posedness and norm inflation type results e.g. [1, 24, 35, 64].

The behavior of solutions in other supercritical models such as the ones arising in fluid and gas dynamics is extremely interesting and not yet well understood. We will not discuss it here.

1.4 Statement of the result

We assert that in dimensions \(5\le d\le 9\) the defocusing (NLS) model (1.1) admits finite time type II (non self similar) blow up solutions arising from \({\mathcal {C}}^\infty \) well localized initial data. The singularity formation is based neither on soliton concentration nor self similar profiles, but on a new front scenario producing a highly oscillatory blow up profile: the leading order dynamics, after renormalization, is given by a type I (self-similar) singularity formation for the compressible Euler equation. The first step of our analysis is to construct \({\mathcal {C}}^\infty \) self-similar solutions to the compressible Euler equations in a suitable range of parameters, which is done in full details in the companion paper [43]. The proof of existence of those solutions involves a non vanishing condition for an explicit constant

$$\begin{aligned} S_\infty (d, \ell )\ne 0, \quad \ell =\frac{4}{p-1}, \end{aligned}$$
(1.4)

which is checked numerically in the range of (pd) considered in [43].

The main result of this paper is the following.

Theorem 1.1

(Existence of energy supercritical type II defocusing blow up) Let

$$\begin{aligned} (d,p)\in \{(5,9),(6,5),(8,3),(9,3)\}, \end{aligned}$$
(1.5)

and let the critical blow up speed be

$$\begin{aligned} r^*(d,\ell )=\frac{\ell +d}{\ell +\sqrt{d}},\quad \ell =\frac{4}{p-1}. \end{aligned}$$
(1.6)

Assume that (1.4) holds for the range (1.5) as is checked numericallyFootnote 1 in [43]. Then there exists a discrete sequence of blow up speeds \((r_k)_{k\ge 1}\) with

$$\begin{aligned} 2<r_k< r^*(d,\ell ), \quad \mathop {\mathrm{lim}}_{k\rightarrow +\infty } r_k=r^*(d,\ell ) \end{aligned}$$

such that any \(k\ge 1\), there exists a finite co-dimensional manifold of smooth initial data \(u_0\in \cap _{m\ge 0} H^m({\mathbb {R}}^d,{\mathbb {C}})\) with spherical symmetry such that the corresponding solution to (1.1) blows up in finite time \(0<T_*<+\infty \) at the center of symmetry with

$$\begin{aligned} \Vert u(t,\cdot )\Vert _{L^\infty }=\frac{c_{p,{k},d}(1+o_{t\rightarrow T_*}(1))}{(T_*-t)^{\frac{1}{p-1}\left( 1+\frac{r_k-2}{r_k}\right) }}, \quad c_{p,r,d}>0. \end{aligned}$$
(1.7)

Comments on the result.

1. Hydrodynamical formulation. The heart of the proof of Theorem 1.1 is a study of (1.1) in its hydrodynamical formulation, i.e. with respect to its phase and modulus variables. The key to our analysis is the identification of an underlying compressible Euler dynamics. The latter arises as a leading order approximation of a “front” like renormalization of the original equation. In this process, the Laplace term applied to the modulusFootnote 2 of the solution is treated perturbatively in the blow up regime. This is one of the key insights of the paper. The approximate Euler dynamics furnishes us with a self-similar solution, which requires very special properties and is constructed in the companion paper [43] and which, in turn, acts as a blow up profile for the original equation. The existence of these blow up profiles is directly related to the restriction on the parameters (1.5) which we discuss in comment 3 below. Let us recall that there is a long history of trying to use the hydrodynamical variables in (NLS) problems and exploit a connection with fluid mechanics, going back to Madelung’s original formulation of quantum mechanics in hydrodynamical variables, [37]. Geometric optics and the hydrodynamical formulation were used to address ill-posedness and norm inflation in the defocusing Schrödinger equations, [1, 24] as well as the study of the semiclassical limit [29]. There is also a recent study of vortex filaments in [3] and its dynamical use of the Hasimoto transform. The scheme of proof of Theorem 1.1 will directly apply to produce the first complete description of singularity formation for the three dimensional compressible Navier–Stokes equation in the companion paper [44].

2. Blow up profile. The blow up profile of Theorem 1.1 is more easily described in terms of the hydrodynamical variables:

$$\begin{aligned} u(t,x)=\rho _{\mathrm{Tot}}(t,x)e^{i\phi (t,x)}. \end{aligned}$$
(1.8)

More precisely, we establish the decomposition

$$\begin{aligned} \left| \begin{array}{ll} \rho _{\mathrm{Tot}}(t,x)=\frac{1}{(T_*-t)^{\frac{1}{p-1}\left( 1+\frac{r-2}{r}\right) }}(\rho _P+\rho )(Z),\\ \phi (t,x)=\frac{1}{(T_*-t)^{\frac{r-2}{r}}}(\Psi _P+\Psi )(Z), \end{array}\right. \quad Z=\frac{x}{(T_*-t)^{\frac{1}{r}}} \end{aligned}$$
(1.9)

and prove the local asymptotic stability

$$\begin{aligned} \mathop {\mathrm{lim}}_{t\rightarrow T_*}\Vert \Psi \Vert _{L^\infty (Z\le 1)}+\Vert \rho \Vert _{L^\infty (Z\le 1)}=0. \end{aligned}$$

Here, the blow up profile \((\rho _P,\Psi _P)\) is, after a suitable transformation, picked among the family of spherically symmetric, smooth and decaying as \(Z\rightarrow +\infty \) self-similar solutions to the compressible Euler equations. The interest in self-similar solutions for the equations of gas dynamics goes back to the pioneering works of Guderley [26] and Sedov [57] (and references therein) who in particular considered converging motion of a compressible gas towards the center of symmetry. However, the rich amount of literature produced since then is concerned with non-smooth self-similar solutions. This is partly due to the physical motivations, e.g. interests in solutions modeling implosion or detonation waves, where self-similar rarefaction or compression is followed by a shock wave (these are self-similar solutions which contain shock discontinuities already present in the data), and, partly due to the fact that, as it turns out, global solutions with the desired behavior at infinity and at the center of symmetry are generically not \({\mathcal {C}}^\infty \). This appears to be a fundamental feature of the self-similar Euler dynamics and, in the language of underlying acoustic geometry, means that generically such solutions are not smooth across the backward light cone (of the acoustical metric associated to the Euler profile) with the vertex at the singularity. The key of our analysis is to find those non-generic \({\mathcal {C}}^\infty \) solutions and to discover that this regularity is an essential element in controlling suitable positivity properties of the associated linearized operator. This is at the heart of the control of the full blow up. A novel contribution of the companion paper [43] is the construction of \({\mathcal {C}}^\infty \) spherically symmetric self-similar solutions to the compressible Euler equations with suitable behavior at infinity and at the center of symmetry for discrete values of the blow up speed parameter r in the vicinity of the limiting blow up speed \(r^*(d,\ell )\) given by (1.6).

3. Restriction on the parameters. There is nothing specific with the choice of parameters (1.5), and we refer to Remark 2.4 for a precise discussion. Two main constraints govern the restriction on the parameters. First of all, a fundamental restriction in order to make the Eulerian regime dominant is the constraint

$$\begin{aligned} r^*(d,\ell )>2\Leftrightarrow \ell <\ell _2(d)=d-2\sqrt{d} \end{aligned}$$
(1.10)

which provides a non empty set of nonlinearities iff

$$\begin{aligned} \ell _2(d)>0\Leftrightarrow d\ge 5. \end{aligned}$$

As a result, the case of dimensions \(d=3,4\) is not amenable to our analysis at this point, and the existence of blow up solutions for \(d=3,4\) remains open. The second restriction concerns the existence of \({\mathcal {C}}^\infty \) smooth blow up profiles with suitable positivity properties of the associated linearized operator, as addressed in [43], see Sect. 2.2 and Remark 2.4 for detailed statements. In particular, a non degeneracy condition \(S_\infty (d,\ell )\ne 0\) for an explicit convergent series is required. An elementary numerical computation to check the condition in the range (1.5).

4. Behavior of Sobolev norms. The conservation of mass and energy imply a uniform \(H^1\) bound on the solution. This can also be checked directly on the leading order representation formulas (1.8), (1.9). For higher Sobolev norms, a computation, see “Appendix D”, shows that the blow up solutions of Theorem 1.1 break scaling, i.e., we can find

$$\begin{aligned} 1<\sigma <s_c=\frac{d}{2}-\frac{2}{p-1} \end{aligned}$$

such that

$$\begin{aligned} \mathop {\mathrm{lim}}_{t\rightarrow T_*}\Vert u(t)\Vert _{H^{\sigma }}=+\infty , \end{aligned}$$

and the critical Sobolev norm \(\Vert u(t,\cdot )\Vert _{H^{s_c}}\) blows up polynomially.

5. Stability of blow up. The blow up profiles of Theorem 1.1 have a finite number of instability directions. Local asymptotic stability in the interior of the backward light (acoustic) cone from the singularity relies on an abstract spectral argument for compact perturbations of maximal accretive operators. Related arguments have been used in the literature for the study of self-similar solutions both in focusing and defocusing regimes, for example [8, 16, 21, 45, 47] for parabolic and [19] for hyperbolic problems. The key to the control of the nonlinear flow in the exterior of the light cone is the propagation of certain weighted scale invariant norms. This generalizes a Lyapunov functional based approach developed in [42]. Counting the precise number of instability directions is an independent problem, disconnected to the nonlinear analysis of the blow up. A natural conjecture is that the number of unstable directions goes to infinity as \(r_k\rightarrow r^*(d, \ell )\).

6. Non spherically symmetric perturbations. We expect that our analysis can be extended to prove the finite codimensional stability of singular dynamics to all perturbations, without the restriction to spherical symmetry. This remains to be done.

7. Oscillatory behavior. The constructed solutions are smooth at the blow up time away from \(x=0\):

$$\begin{aligned} \forall R>0, \ \ \mathop {\mathrm{lim}}_{t\rightarrow T_*}u(t,x)= u^*(x) \ \ \text{ in }\ \ H^k(|x|>R), \ \ k\in {\mathbb {N}}. \end{aligned}$$
(1.11)

As in the cases for blow up problems in the focusing setting, see e.g. [40], the profile outside the blow up point has a universal behavior when approaching the singularity

$$\begin{aligned} u^*(x)=c_P(1+o_{|x|\rightarrow 0}(1))\frac{e^{i\frac{c_\Psi }{|x|^{r-2}}}}{|x|^{\frac{2(r-1)}{p-1}}}, \ \ c_P\ne 0. \end{aligned}$$
(1.12)

What is unusual, and together with potential non-genericity perhaps responsible for difficulties in numerical detection of the blow up phenomena, is the highly oscillatory behavior. This appears to be a deep consequence of the structure of the self-similar solution to the compressible Euler equation and the coupling of phase and modulus variable in the blow up regime, generating an anomalous Euler scaling. The heart of our analysis is to show that after passing to the suitable renormalized variables provided by the front, the highly oscillatory behavior (1.12) becomes regular near the singularity and can be controlled with the monotonicity estimates of energy type, without appealing to Fourier analysis.

8. On the role of the defocusing nonlinearity. The existence of self similar solutions to the energy supercritical (NLS) decaying at infinity is expected to hold in the focusing case, like for the heat equation [15]. In the defocusing case, such solutions are easily ruled out for the heat equation using the maximum principle, and their non existence is an open problem for the defocusing NLS, we refer to [33] for further discussion in the case of the wave equation. A fundamental observation is that in a suitable range of parameters, the semiclassical Euler limit provides admissible approximate blow up profiles for the defocusing NLS. The fact that our range of parameters is energy supercritical can be seen directly on the constraint (1.10):

$$\begin{aligned} r^*(d, \ell )>2\Leftrightarrow & {} \frac{4}{p-1}=\ell <d-2\sqrt{d}\\\Leftrightarrow & {} p>1+\frac{4}{d-2\sqrt{d}}> p_c=1+\frac{4}{d-2}. \end{aligned}$$

In other words, the existence of suitable blow up profiles given by Euler (\(r<r^*(d, \ell )\)) combined with the constraint that the Euler regime dominates (\(r>2\)) forces an energy supercritical range of parameters.

The paper is organized as follows. In Sect. 2, we present the “front” renormalization of the flow which makes the Euler dynamics dominant, and recall all necessary facts about the corresponding self similar profile built in [43]. Theorem 1.1 reduces to building a global in time non vanishing solution to the renormalized flow (2.25) written in hydrodynamical variables. In Sect. 2.4 we detail the strategy of the proof. In Sect. 3, we introduce the functional setting related to maximal accretivity (modulo a compact perturbation) of the corresponding linear operator which leads to a statement of exponential decay in a neighborhood of the light cone for the space of solutions (modulo an a priori control of a finite dimensional manifold corresponding to the unstable directions.) In Sect. 4, we describe our set of initial data and the set of bootstrap assumptions which govern the analysis. In Sects. 567, we close the control of weighted Sobolev norms and the associated pointwise bounds. In Sect. 8, we close the exponential decay of low Sobolev norms by relying on spectral estimates and finite speed of propagation arguments.

1.5 Notations

The bracket

$$\begin{aligned} \langle r\rangle =\sqrt{1+r^2}. \end{aligned}$$

The weighted scalar product for a given measure g:

$$\begin{aligned} (u,v)_g=\int _{{\mathbb {R}}^d} u{\overline{v}}g\,dx. \end{aligned}$$
(1.13)

The integer part of \(x\in {\mathbb {R}}\)

$$\begin{aligned} x\le [x]<x+1, \ \ [x]\in {\mathbb {Z}}. \end{aligned}$$

The infinitesimal generator of dilations

$$\begin{aligned} \Lambda =y\cdot \nabla . \end{aligned}$$

2 Front renormalization, blow up profile and strategy of the proof

In this section we introduce the hydrodynamical variables to study (1.1) and the associated renormalization procedure which makes the compressible Euler structure dominant. We collect from [43] the main facts about the existence of smooth spherically symmetric self-similar solutions to the compressible Euler equations which will serve as blow up profiles.

2.1 Hydrodynamical formulation and front renormalization

We begin by establishing a link between the Eq. (1.1) and the compressible Euler equations. For non vanishing solutions, we write the equivalent hydrodynamical formulation in phase and modulus variables:

$$\begin{aligned} u(t,x)=\sqrt{\rho }({{\tilde{t}}},x) e^{i\psi ({{\tilde{t}}},x)},\qquad {{\tilde{t}}}=2t. \end{aligned}$$

Equation (1.1) becomes a system

$$\begin{aligned} \left| \begin{array}{l} \partial _{{{\tilde{t}}}}\rho +\rho \Delta \psi +\nabla \rho \cdot \nabla \psi =0,\\ \partial _{{{\tilde{t}}}}\psi +\frac{1}{2} |\nabla \psi |^2+\frac{1}{2}\rho ^{\frac{p-1}{2}}=\frac{1}{2} \frac{\Delta \sqrt{\rho }}{\sqrt{\rho }}. \end{array}\right. \end{aligned}$$

This is precisely the compressible Euler (potential flow) equations (the second equation is the Bernoulli equation) for the density \(\rho \), velocity \(\nabla \psi \), the classical pressure

$$\begin{aligned} P=\frac{p-1}{2(p+1)}\rho ^{\frac{p+1}{2}} \end{aligned}$$

and the quantum stress tensor

$$\begin{aligned} {\mathcal {Q}}=\frac{1}{2} \Delta \rho I -\frac{1}{2} \frac{\nabla \rho \otimes \nabla \rho }{\rho }, \end{aligned}$$

so that

$$\begin{aligned} \left| \begin{array}{l} \partial _{{{\tilde{t}}}}\rho +\mathrm{div \;}(\rho \cdot \nabla \psi )=0,\\ \rho \partial _{{{\tilde{t}}}}\nabla \psi + \rho \nabla \psi \cdot \nabla \nabla \psi +\nabla P=\mathrm{div \;}{\mathcal {Q}}. \end{array}\right. \end{aligned}$$
(2.1)

Below, we will show that passing to self-similar variables, the above system admits an additional front renormalization which damps the quantum stress term and therefore possesses approximate stationary solutions which, in turn, are self-similar solutions of the classical Euler equations. For convenience, we work in slightly different variables (using density squared in place of density, for instance). The correspondence between the systems derived below and the compressible Euler equations, along the lines of (2.1), will hold at every step. The explicit identification of the final approximate system (2.11) with the corresponding system representing self-similar solutions of the Euler equations is done in “Appendix A”.

The standard self-similar renormalization

$$\begin{aligned} u(t,x)=\frac{1}{\lambda (t)^{\frac{2}{p-1}}}v({\tau },y)e^{i\gamma }, \ \ y=\frac{x}{\lambda }, \end{aligned}$$

where we freeze the scaling parameter at the self-similar scale

$$\begin{aligned} \frac{d\tau }{dt}=\frac{1}{\lambda ^2}, \ \ y=\frac{x}{\lambda (t)}, \ \ -\frac{\lambda _\tau }{\lambda }=\frac{1}{2}, \end{aligned}$$

then (1.1) becomes

$$\begin{aligned} i\partial _\tau v+\Delta v-\gamma _\tau v-i\frac{\lambda _\tau }{\lambda }\left( \frac{2}{p-1}v+ \Lambda v\right) -v|v|^{p-1}=0. \end{aligned}$$
(2.2)

In the defocusing case, (2.2) has no known decaying type I self similar stationary solution, or type II soliton like solutions, [42], but, it turns out, that it admits approximate front like solutions. Their existence relies on a specific phase and modulus coupling and anomalous scaling. We introduce the parameters

$$\begin{aligned} \left| \begin{array}{l} r=\frac{2}{1-{\mathscr {e}}}, \ \ 0<{\mathscr {e}} <1,\\ \mu =\frac{1}{r}=\frac{1-{\mathscr {e}}}{2}\\ \ell =\frac{4}{p-1} \end{array}\right. \end{aligned}$$
(2.3)

and claim:

Lemma 2.1

(Front renormalization of the self similar flow) Define geometric parameters

$$\begin{aligned} -\frac{\lambda _\tau }{\lambda }=\frac{1}{2}, \ \ \frac{b_\tau }{b}=-{\mathscr {e}}, \ \ \gamma _\tau =-\frac{1}{b}, \ \ \frac{d\tau }{dt}=\frac{1}{\lambda ^2} \end{aligned}$$
(2.4)

and introduce the renormalization

$$\begin{aligned} u(t,x)=\frac{1}{\lambda (t)^{\frac{2}{p-1}}}v({\tau },y)e^{i\gamma }, \ \ y=\frac{x}{\lambda } \end{aligned}$$

with the phase and modulus

$$\begin{aligned} \left| \begin{array}{l} v=we^{i\phi },\\ w(\tau ,y)=\frac{1}{(\sqrt{b})^{\frac{2}{p-1}}}\rho _{\mathrm{Tot}}(\tau ,Z)\in {\mathbb {R}}^*_+,\\ \phi (\tau ,y)=\frac{1}{b}\Psi _{\mathrm{Tot}}(\tau ,Z),\\ Z={|y|}\sqrt{b}, \end{array}\right. \end{aligned}$$

In these variables (1.1) becomes, on \([\tau _0,+\infty )\):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho _{\mathrm{Tot}}=-\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}-\frac{\mu \ell (r-1)}{2}\rho _{\mathrm{Tot}}-\left( 2\partial _Z\Psi _{\mathrm{Tot}}+\mu Z\right) \partial _Z\rho _{\mathrm{Tot}},\\ \rho _{\mathrm{Tot}}\partial _\tau \Psi _{\mathrm{Tot}}=b^2\Delta \rho _{\mathrm{Tot}}\\ \quad -\left[ |\nabla \Psi _{\mathrm{Tot}}|^2+\mu (r-2)\Psi _{\mathrm{Tot}}-1+\mu \Lambda \Psi _{\mathrm{Tot}}+\rho _{\mathrm{Tot}}^{p-1}\right] \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$
(2.5)

Remark 2.2

Since from (2.4) we have frozen the scaling in its selfsimilar law, the lifetime of the solution in original variables is \(T_*=e^{-\tau _0}\), see also (4.2).

Proof

Starting from (2.2), we define a polar decomposition

$$\begin{aligned} v=we^{i\phi } \end{aligned}$$

so that

$$\begin{aligned} v'=(w'+i\phi 'w)e^{i\phi }, \ \ v''=w''-|\phi '|^2w+2i\phi 'w'+i\phi '' w \end{aligned}$$

and

$$\begin{aligned} 0= & {} i\partial _\tau w+\Delta w+\left( -\partial _\tau \phi -|\nabla \phi |^2-\gamma _\tau +\frac{\lambda _\tau }{\lambda }y\cdot \nabla \phi \right) w \nonumber \\&+ i\left( \Delta \phi -\frac{2}{p-1}\frac{\lambda _\tau }{\lambda }\right) w+i\left( 2\nabla \phi -\frac{\lambda _\tau }{\lambda } y\right) \cdot \nabla w-w|w|^{p-1}.\nonumber \\ \end{aligned}$$
(2.6)

Separating the real and imaginary parts yields the self-similar equation (2.2):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau w=-\left( \Delta \phi +\frac{1}{p-1}\right) w-\left( 2\frac{\partial _y\phi }{{|y|}}+\frac{1}{2}\right) \Lambda w,\\ w\partial _\tau \phi =\Delta w+\left( -|\nabla \phi |^2-\gamma _\tau -\frac{1}{2}\Lambda \phi \right) w -w|w|^{p-1}. \end{array}\right. \end{aligned}$$
(2.7)

We now renormalize according to

$$\begin{aligned} w(\tau ,y)=\frac{1}{(\sqrt{b})^{\frac{2}{p-1}}}\rho _{\mathrm{Tot}}(\tau ,Z)\in {\mathbb {R}}^*_+, \ \ \phi (\tau ,y)=\frac{1}{b}\Psi _{\mathrm{Tot}}(\tau ,Z) \ \ Z={|y|}\sqrt{b} \end{aligned}$$

with a fixed choice of parameters in the modulation equations

$$\begin{aligned} \frac{b_\tau }{b}=-{{\mathscr {e}}}, \ \ \gamma _\tau =-\frac{1}{b}, \ \ 0<{{\mathscr {e}}}<1 \end{aligned}$$

which transforms (2.7) into

$$\begin{aligned} \left| \begin{array}{ll}\partial _\tau \rho _{\mathrm{Tot}}=-\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}-\frac{{{\mathscr {e}}}+1}{p-1}\rho _{\mathrm{Tot}}-\left( 2\partial _Z\Psi _{\mathrm{Tot}}+\frac{1-{{\mathscr {e}}}}{2}Z\right) \partial _Z\rho _{\mathrm{Tot}},\\ \rho _{\mathrm{Tot}}\partial _\tau \Psi _{\mathrm{Tot}}=b^2\Delta \rho _{\mathrm{Tot}}\\ \quad -\left[ |\nabla \Psi _{\mathrm{Tot}}|^2+{{\mathscr {e}}}\Psi _{\mathrm{Tot}}-1+\frac{1}{2}(1-{{\mathscr {e}}})\Lambda \Psi _{\mathrm{Tot}}+\rho _{\mathrm{Tot}}^{p-1}\right] \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$

We now compute from (2.3):

$$\begin{aligned} \left| \begin{array}{l} \frac{\mu \ell (r-1)}{2}=\frac{2}{p-1}(1-\mu )=\frac{1+{{\mathscr {e}}}}{p-1}\\ \mu (r-2)=1-(1-{{\mathscr {e}}})={{\mathscr {e}}} \end{array}\right. , \end{aligned}$$

and (2.5) is proved. \(\square \)

2.2 Blow up profile and Emden transform

We recall in this section the main results of [43].

Emden transform. A stationary solution \((\rho _P,\Psi _P)\) to (2.5) in the limiting Eulerian regime \(b=0\) satisfies the profile equation

$$\begin{aligned} \left| \begin{array}{ll} |\nabla \Psi _P|^2+\rho _P^{p-1}+\mu (r-2)\Psi _P+\mu \Lambda \Psi _P=1,\\ \Delta \Psi _P+\frac{\mu \ell (r-1)}{2}+\left( 2\partial _Z\Psi _P+\mu Z\right) \frac{\partial _Z\rho _P}{\rho _P}=0. \end{array}\right. \end{aligned}$$
(2.8)

We supplement it with the boundary conditions:

$$\begin{aligned} \left| \begin{array}{ll} \rho _P(0)=1, \ \ \Psi _P(0)=0,\\ \rho _P({Z})\rightarrow 0,\ \Psi _P({Z})\rightarrow \frac{1}{{\mathscr {e}}} \ \ {\text{ a }s}\ \ {Z}\rightarrow \infty . \end{array}\right. \end{aligned}$$
(2.9)

We now show that the system (2.8), (2.9) is equivalent to the corresponding system of equations describing self-similar solutions of the Euler equations. We define the Emden variables:

$$\begin{aligned} \left| \begin{array}{lll} \phi =\frac{\mu }{2}\sqrt{\ell }, \ \ p-1=\frac{4}{\ell },\\ Q=\rho _P^{p-1}=\frac{1}{M^2}, \ \ \frac{1}{M}=\phi Z \sigma ,\\ \frac{\Psi '_P}{Z}=-\frac{\mu }{2}w, \end{array}\right. \ \ {\varkappa }=\mathrm{log}Z, \end{aligned}$$
(2.10)

then (2.8) is mapped onto

$$\begin{aligned} \left| \begin{array}{ll} (w-1)w'+\ell \sigma \sigma '+(w^2-rw+\ell \sigma ^2)=0,\\ \frac{\sigma }{\ell }w'+(w-1)\sigma '+\sigma \left[ w\left( \frac{d}{\ell }+1\right) -r\right] =0, \end{array} \right. \end{aligned}$$
(2.11)

or equivalently

$$\begin{aligned} \left| \begin{array}{ll}a_1w' +b_1\sigma '+d_1=0,\\ a_2 w'+b_2\sigma '+d_2=0 \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} \left| \begin{array}{ll} a_1=w-1, \ \ b_1=\ell \sigma , \ \ d_1=w^2-rw+\ell \sigma ^2,\\ a_2=\frac{\sigma }{\ell }, \ \ b_2=w-1, \ \ d_2=\sigma \left[ \left( 1+\frac{d}{\ell }\right) w-r\right] . \end{array}\right. \end{aligned}$$
(2.12)

The system (2.11) is exactly the one describing spherically symmetric self-similar solutions to the compressible Euler equation, [57] (and the references therein). For an explicit derivation see “Appendix A”. It is analyzed in [43], following pioneering work of Guderley, Sedov and others.

Let

$$\begin{aligned} w_e=\frac{\ell (r-1)}{d} \end{aligned}$$
(2.13)

and the determinants

$$\begin{aligned} \left| \begin{array}{lll} \Delta =a_1b_2-b_1a_2=(w-1)^2-\sigma ^2,\\ \Delta _1=-b_1d_2+b_2d_1=w(w-1)(w-r)-d(w-w_e)\sigma ^2,\\ \Delta _2=d_2a_1-d_1a_2=\frac{\sigma }{\ell }\left[ (\ell +d-1)w^2-w(\ell +d+\ell r-r)+\ell r-\ell \sigma ^2\right] \end{array}\right. \nonumber \\ \end{aligned}$$
(2.14)

then

$$\begin{aligned} w'=-\frac{\Delta _1}{\Delta }, \ \ \sigma '=-\frac{\Delta _2}{\Delta }, \ \ \frac{dw}{d\sigma }=\frac{\Delta _1}{\Delta _2}. \end{aligned}$$
(2.15)

Solution curves \(w=w(\sigma )\) of the above system can be examined through its phase portrait in the \((\sigma ,w)\) plane.

Critical points and admissible profile. The shape of the phase portrait depends crucially on the polynomials \(\Delta \), \(\Delta _{1}\), \(\Delta _2\) and the parameters \((r,d,\ell )\), and we refer to [43] for a complete description. In particular, three critical points play a distinguished role in the analysis:

  • the \(P_6\) unstable point which corresponds to a point at infinity \((\sigma =+\infty , w=w_e)\), which in original variables corresponds to the smooth solution coming out of the origin \(Z=0\),

  • the \(P_4\) stable point \((\sigma =0, w=0)\) which corresponds to selfsimilar decay as \(Z\rightarrow +\infty \),

  • the \(P_2\) stable point which is a solution to the triple point equation

    $$\begin{aligned} \Delta (P_2)=\Delta _1(P_2)=\Delta _2(P_2) =0. \end{aligned}$$
    (2.16)

A classical analysis of the phase portrait reveals that in a suitable regime of parameters, there is a unique solution coming out of \(P_6\) with the normalization

$$\begin{aligned} \rho _P(0)=1, \ \ \Psi _P(0)=0 \end{aligned}$$
(2.17)

at \({Z}=0\), which is also \({{\mathcal {C}}}^\infty \) in the vicinity of \({Z}=0\), and it must be attracted into \(P_2\). This solution can be continued beyond \(P_2\) by gluing it to a member of the 1-parameter family of curves that join \(P_2\) to the selfsimilar decay \(P_4\) as \(Z\rightarrow +\infty \).

The above procedure produces a curve which is \({{\mathcal {C}}}^\infty \) everywhere except at \(P_2\) where it generically experiences an unavoidable discontinuity of high derivatives, except for discrete values of the speed r. The following structural proposition on the blow up profile is proved in the companion paper [43].

Theorem 2.3

(Existence and asymptotics of a \({\mathcal {C}}^\infty \) profile, [43]) Let

$$\begin{aligned} (d,p)\in \{(5,9),(6,5),(8,3),(9,3)\} \end{aligned}$$

and recall (1.6). Then there exists a sequence \((r_k)_{k\ge 1}\) with

$$\begin{aligned} \mathop {\mathrm{lim}}_{k\rightarrow \infty } r_k=r^*(d,\ell ), \quad r_k<r^*(d,\ell ) \end{aligned}$$
(2.18)

such that for all \(k\ge 1\), the following holds:

  1. 1.

    Existence of a smooth profile at the origin: the unique radially symmetric solution to (2.8) with Cauchy data at the origin (2.9) reaches in finite time \(Z_2>0\) the point \(P_2\).

  2. 2.

    Passing through \(P_2\): the solution passes through \(P_2\) with \({\mathcal {C}}^\infty \) regularity.

  3. 3.

    Large Z asymptotic: the solution admits the asymptotics as \(Z\rightarrow +\infty \):

    $$\begin{aligned} \left| \begin{array}{l} w(Z)=\frac{c_w}{Z^r}\left( 1+O\left( \frac{1}{Z^{ r}}\right) \right) \\ \sigma (Z)=\frac{c_\sigma }{Z^r}\left( 1+O\left( \frac{1}{Z^{r}}\right) \right) \\ \end{array}\right. \end{aligned}$$
    (2.19)

    or equivalently

    $$\begin{aligned} \left| \begin{array}{ll} Q(Z)=\rho _P^{p-1}(Z)=\frac{c_P^{p-1}}{Z^{2(r-1)}}\left( 1+O\left( \frac{1}{Z^r}\right) \right) , \\ \Psi _P(Z)=\frac{1}{{\mathscr {e}}}+\frac{c_\Psi }{Z^{r-2}}\left( 1+O\left( \frac{1}{Z^r}\right) \right) \end{array}\right. \end{aligned}$$
    (2.20)

    with non zero constants \(c_\sigma ,c_P\). Similar asymptotics hold for all higher order derivatives.

  4. 4.

    Non vanishing: there holds

    $$\begin{aligned} \forall Z\ge 0, \quad \rho _P>0. \end{aligned}$$
  5. 5.

    Strict positivity inside the light cone: there exists \(c=c(d,\ell ,r)>0\) such that

    $$\begin{aligned} \forall 0\le Z\le Z_2, \quad \left| \begin{array}{l} (1-w-\Lambda w)^2-{(\sigma +\Lambda \sigma )^2}>c\\ 1-w- \Lambda w-\frac{(1-w){(\sigma +\Lambda \sigma )}}{\sigma }> c. \end{array}\right. \end{aligned}$$
    (2.21)
  6. 6.

    Strict positivity outside the light cone:

    $$\begin{aligned} \exists c=c_{d,\ell ,r}>0, \quad \forall Z\ge Z_2, \quad \left| \begin{array}{l} (1-w-\Lambda w)^2- {(\sigma +\Lambda \sigma )^2}>c,\\ 1-w-\Lambda w>c. \end{array}\right. \end{aligned}$$
    (2.22)

Remark 2.4

(Restriction on the parameters) The proof of Theorem 2.3 requires the non degeneracy of an explicit series \(S_\infty (d,\ell )\ne 0\) which is numerically checked in [43] in the range (1.5). The positivity properties (2.21), (2.22) are checked analytically in [43] and will be fundamental for the well-posedness of the linearized flow inside the light cone, and the control of global Sobolev norms outside the light cone. Let us insist that the restriction on parameters relies on the intersection of the conditions (1.10), \(S_\infty (d,\ell )\ne 0\) and (2.21), (2.22). The range (1.5) is just an example where this holds, but a larger range of parameters can be directly extracted from [43], and the conclusion of Theorem 1.1 would follow. In particular, since we are working with non vanishing solutions, the fact that the non linearity is an odd integer can be relaxed as in [44], hence providing an open range of parameters. Determining the exact range of validity of parameters for which Theorem 1.1 holds remains open.

Remark 2.5

The strict positivity property (2.21) inside the light cone will play a distinguished role in the analysis of the linearized of the operator and the derivation of the spectral gap which is the key to decay, see Proposition 3.10. Together with the strict positivity (2.22) outside the light cone, it will also allow us to derive energy bounds at high regularity, see Proposition 7.1.

From now on and for the rest of this paper, we assume (1.5). We observe from direct check that there holds:

$$\begin{aligned} r^*(\ell )=\frac{d+\ell }{\ell +\sqrt{d}}>2\Leftrightarrow \ell <d-2\sqrt{d}=\ell _2(d). \end{aligned}$$

Recalling (2.3), we may therefore assume from (2.18) that the blow speed \(r=r_k\) satisfies

$$\begin{aligned} r>2\Leftrightarrow {{\mathscr {e}}}=\frac{r-2}{r}>0. \end{aligned}$$

2.3 Linearization of the renormalized flow

We look for u solution to (1.1) and proceed to the decomposition of Lemma 2.1. We are left with finding a global, in self similar time \(\tau \in [\tau _0,+\infty )\), solution to (2.5):

$$\begin{aligned} \left| \begin{array}{ll}\partial _\tau \rho _{\mathrm{Tot}}=-\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}-\frac{\mu \ell (r-1)}{2}\rho _{\mathrm{Tot}}-\left( 2\partial _Z\Psi _{\mathrm{Tot}}+\mu Z\right) \partial _Z\rho _{\mathrm{Tot}},\\ \rho _{\mathrm{Tot}}\partial _\tau \Psi _{\mathrm{Tot}}=b^2\Delta \rho _{\mathrm{Tot}}\\ \qquad \qquad -\left[ |\nabla \Psi _{\mathrm{Tot}}|^2+\mu (r-2)\Psi _{\mathrm{Tot}}-1+\mu \Lambda \Psi _{\mathrm{Tot}}+\rho _{\mathrm{Tot}}^{p-1}\right] \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$
(2.23)

with non vanishing density \(\rho _{\mathrm{Tot}}>0\). We define

$$\begin{aligned} \left| \begin{array}{ll} H_2=\mu +2\frac{\Psi _{P}'}{Z}=\mu (1-w),\\ H_1=-\left( \Delta \Psi _{P}+\frac{\mu \ell (r-1)}{2}\right) =H_2\frac{\Lambda \rho _{P}}{\rho _{P}}=\frac{\mu \ell }{2}(1-w)\left[ 1+\frac{\Lambda \sigma }{\sigma }\right] . \end{array}\right. \end{aligned}$$
(2.24)

We linearize

$$\begin{aligned} \rho _{\mathrm{Tot}}=\rho _P+\rho , \quad \Psi _{\mathrm{Tot}}=\Psi _P+\Psi \end{aligned}$$

and compute, using the profile Eq. (2.8), for the first equation:

$$\begin{aligned} \partial _\tau \rho= & {} -(\rho _P+\rho )\Delta (\Psi _P+\Psi )-\frac{\mu \ell (r-1)}{2}(\rho _P+\rho )\\&-(2\partial _Z\Psi _P+\mu Z+2\partial _Z\Psi )(\partial _Z\rho _P+\partial _Z\rho )\\= & {} -\rho _{\mathrm{Tot}}\Delta \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi +H_1\rho -H_2\Lambda \rho \end{aligned}$$

and for the second one:

$$\begin{aligned} \rho _{\mathrm{Tot}}\partial _\tau \Psi= & {} b^2\Delta \rho _{\mathrm{Tot}}-\rho _{\mathrm{Tot}}\Big \{|\nabla \Psi _P|^2+2\nabla \Psi _P\cdot \nabla \Psi +|\nabla \Psi |^2- 1+\mu (r-2)\Psi _P\\&+\mu (r-2)\Psi +\mu (\Lambda \Psi _P+\Lambda \Psi )\\&+(\rho _P+\rho )^{p-1}\Big \}\\= & {} b^2\Delta \rho _{\mathrm{Tot}}-\rho _{\mathrm{Tot}}\left\{ 2\nabla \Psi _P\cdot \nabla \Psi +\mu \Lambda \Psi +\mu (r-2)\Psi \right. \\&\left. +|\nabla \Psi |^2+(\rho _P+\rho )^{p-1}-\rho _P^{p-1}\right\} \\= & {} b^2\Delta \rho _{\mathrm{Tot}}-\rho _{\mathrm{Tot}}\left\{ H_2\Lambda \Psi +\mu (r-2)\Psi \right. \\&\left. +|\nabla \Psi |^2+(p-1)\rho _P^{p-2}\rho +\text {NL}(\rho )\right\} \end{aligned}$$

with

$$\begin{aligned} \text {NL}(\rho )=(\rho _P+\rho )^{p-1}-\rho _P^{p-1}-(p-1)\rho _P^{p-2}\rho . \end{aligned}$$

We arrive at the exact (nonlinear) linearized flow

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho =H_1\rho -H_2\Lambda \rho -\rho _{\mathrm{Tot}}\Delta \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ,\\ \partial _\tau \Psi =b^2\frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\left\{ H_2\Lambda \Psi +\mu (r-2)\Psi +|\nabla \Psi |^2\right. \\ \quad \left. +(p-1)\rho _P^{p-2}\rho +\text {NL}(\rho )\right\} . \end{array}\right. \end{aligned}$$
(2.25)

Theorem 1.1 is therefore equivalent to exhibiting a finite co-dimensional manifold of smooth well localized initial data leading to global, in renormalized \(\tau \)-time, solutions to (2.25).

2.4 Strategy of the proof

We now explain the strategy of the proof of Theorem 1.1.

Step 1 Wave equation and propagator estimate. After the change of variables \(\Phi =\rho _P\Psi \), we may schematically rewrite the linearized flow (2.25) in the form

$$\begin{aligned} \partial _\tau X={\mathcal {M}} X+\text {NL}(X)-b^2\left| \begin{array}{ll}0\\ \Delta (\rho _P+\rho )\end{array}\right. \end{aligned}$$
(2.26)

with

$$\begin{aligned} X= \left| \begin{array}{ll}\rho \\ \Phi \end{array}\right. , \ \ {\mathcal {M}}=\left( \begin{array}{ll}H_1-H_2\Lambda \&-\Delta +H_3p-1)Q-H_2\Lambda&H_1-\mu (r-2)\end{array}\right) , \end{aligned}$$
(2.27)

where \(Q,H_1,H_2,H_3\) are explicit potentials generated by the profile \(\rho _P,\Psi _P\). During the first step the \(b^2\Delta \) term is treated perturbatively. We commute the equation with the powers of the laplacian \(\Delta ^k\) and obtain for \(X_k=\Delta ^kX\)

$$\begin{aligned} \partial _\tau X_k={\mathcal {M}}_k X+\text {NL}_k(X). \end{aligned}$$
(2.28)

We then show that, provided k is large enough, \({\mathcal {M}}_k\) is a finite rank perturbation of a maximally dissipative operator with a spectral gap \(\delta >0\). The topology in which maximal accretivity is established depends on the properties of the wave equationFootnote 3 encoded in (2.28) and is based on weighted Sobolev norms with weights vanishing on the light cone corresponding to the point P2 of the profile. Indeed, the principal part of the wave equation is roughly of the form

$$\begin{aligned} \partial _\tau ^2\rho -D(Z)\partial _Z^2\rho , \end{aligned}$$

where the weight D(Z) vanishes on the light cone \(Z=Z_2\) corresponding to the \({P_2}\) point. The corresponding propagation estimates for the wave equation produce an priori control of the solution in the interior of the light cone \(Z<Z_2\), modulo an a priori control of a finite number of directions corresponding to non positive eigenvalues of \({\mathcal {M}}_k\). An essential structural fact of this step is the \({\mathcal {C}}^\infty \) regularity of the profile. Indeed, we claim that for a generic non \({\mathcal {C}}^\infty \) solution at P2, the number of derivatives required to show accretivity of the linearized operator is always strictly greater than the regularity of the profile at \({P_2}\). As a result such profiles may be completely unstable and are not amenable to our analysis. The \({\mathcal {C}}^\infty \) regularity obtained in [43] is therefore absolutely fundamental. The analytic properties leading to the maximality of the linearized operator will be consequences of (2.21), (2.22). We note that the coercivity constant in (2.21) degenerates as \(r\rightarrow r^*\), and the number of derivatives needed for accretivity is inversely proportional to this constant. This is a manifestation of a quasilinear effect which is new for NLS: the problem sees a scaling which depends on the chosen self similar profile.

Step 2 Extension slightly beyond the light cone. Exponential decay estimates provided in the first step yield control in the interior of the light cone \(Z< Z_2\) only. It turns out that the analysis of the first step can be made more robust and extendedFootnote 4 slightly beyond the light cone, all the way to a spacelike hypersurface \(Z=Z_2+a\), \(0<a\ll 1\), even though it is complicated by the dependence of the underlying wave equation on variable coefficients or, equivalently, on non constancy of the Q(Z) term in (2.27). We can revisit the first step by producing a new maximal accretivity structure for a norm which does not generate in the zone \(Z<Z_2+a\), \(0<a\ll 1\). The argument relies on a new generalized monotonicity formula. The corresponding propagation estimates recovers exponential decay in the extended zone \(Z<Z_2+a\). Once decay has been obtained strictly beyond the light cone, a simple finite speed of propagation argument allows us to propagate decay to any compact set \(Z<Z_0\), \(Z_0\gg 1\).

Step 3 Loss of derivatives. The decay obtained in step 2 relies on energy estimates compatible with the wave propagation and the Eulerian structure of approximation. The full evolution however is that of the Schrödinger equation and contains the \(b^2\Delta \) term on the right hand side of (2.26). Such a term leads to an unavoidable loss of one derivative. However, this loss comes with a \(b^2\) smallness in front. We then argue as follows. We pick a large enough regularity level \(k_m=k_m(r,d)\gg 2k_0\), where \(k_0\) is the power of the laplacian used for commutation in step 2, and derive a global Schrödinger like energy identity on the full flow (2.25). The choice of phase and modulus as basic variables turns the equation quasilinear and makes this identity rather complicated and unfamiliar. An essential difficulty, which is deeply related to step 2, is that at the highest level of derivatives, the non trivial space dependence of the profile measured by \(Q(Z)=\rho _P^{p-1}(Z)\) in (2.27) produces a coupling term and a non trivial quadratic form. The condition (2.22) implies that the corresponding quadratic form is definite positive for \(k_m\) large enough.

Step 4 Closing estimates. As explained above, we work with a linearized nonlinear equation, i.e., obtained after subtracting off the profile, written in terms of the phase and modulus unknowns \((\Psi , \rho )\), in renormalized self-similar variables \((\tau , Z)\), where the singularity corresponds to \((\tau =\infty , Z=0)\), a special light cone is \((\tau , Z=Z_2)\) and where in the original variables (tr) the region \(r\ge 1\) corresponds to \(Z\ge e^{\mu \tau }\).

First, outside the singularity \(r\ge 1\), we modify the profile by strengthening its decay to make it rapidly decaying and of finite energy. Relative to the self-similar variables this modification happens at \(Z\sim e^{\mu \tau }\), far from the singularity, and as a result is harmless. Then, we run two sets of estimates. First, we employ wave propagation like estimates which go initially just slightly beyond the special light cone and then extend to any compact set in Z. These estimates are carried out at a sufficiently high level of regularity with \(\sim 2k_0\) derivatives. The number \(k_0\) emerges from the linear theory and is determined by the (conditional) positivity of a certain quadratic form responsible for maximal accretivity.

Then, we couple these estimates to global Schrödinger like estimates which take into account previously ignored \(b^2\Delta \) and take care of global control. These estimates are carried out at all levels of regularity up to \(k_m\) derivatives with \(k_m\gg k_0\). They are carefully designed weighted \(L^2\) type estimates. The weights depend on the number of derivatives k: at first, their strength grows with k but by the time we reach the highest level of regularity \(k_m\) the weight function is identically \(=1\). The latter has to do with a well-known fact that even for a linear Schrödinger equation, use of weights leads to a derivative loss (\(\Delta \) is not self-adjoint on a weighted \(L^2\) space.) Therefore, our highest derivative norm should correspond to an unweighted \(L^2\) estimate. Of course, this last estimate also sees a positivity condition (2.22) responsible for the coercivity of an appearing quadratic form.

These global weighted \(L^2\) bounds then allow us to prove pointwise bounds for the solution and its derivatives which, in turn, allow us to control nonlinear terms. The obtained sets of weighted \(L^\infty \) bounds on derivatives recover in particular the non vanishing assumption required of the solution. We should note that while all the local (in Z) norms decay exponentially in \(\tau \), the global norms are merely bounded. In the original (tr) variables this means that the perturbation decays inside and slightly beyond the backward light cone from the singular point but does not decay away from the singularity. This is, of course, entirely consistent with the global conservation of energy for NLS.

The whole proof proceeds via a bootstrap argument which also involves a Brouwer type argument to deal with unstable modes, if any, arising in linear theory of step 1. This is what produces a finite co-dimension manifold of admissible data.

3 Linear theory slightly beyond the light cone

Our aim in this section is to study the linearized problem (2.25) for the exact Euler problem \(b=0\). We in particular aim at setting up the suitable functional framework in order to apply classical propagator estimates which will yield exponential decay on compact sets \(Z\lesssim 1\) modulo the control of a finite number of unstable directions.

3.1 Growth bounds for dissipative operators

We start this section by recalling classical facts about unbounded operators and their semigroups. Let \((H,\langle \cdot ,\cdot \rangle )\) be a hermitian Hilbert space and A be a closed operator with a dense domain D(A). We recall the definition of the adjoint operator \(A^*\): let

$$\begin{aligned} D(A^*)=\{X\in H, \ \ {\tilde{X}}\in D(A)\mapsto & {} \langle X, A{\tilde{X}} \rangle \\&\text{ extends } \text{ as } \text{ a } \text{ bounded } \text{ functional } \text{ on }\ \ H\}, \end{aligned}$$

then \(A^*X\) is given by the Riesz theorem as the unique element of H such that

$$\begin{aligned} \forall {\tilde{X}}\in D(A), \ \ \langle A^*X,{\tilde{X}}\rangle =\langle X,A{\tilde{X}}\rangle . \end{aligned}$$
(3.1)

Let \(\sigma (A)\) denote the spectrum of A, i.e., the complement of the resolvent set. We recall the following classical lemma.

Lemma 3.1

(Properties of maximal dissipative operators, [56] p. 49) Let A be a maximal dissipative operator on a Hilbert space H with domain D(A), then:

  1. (i)

    A is closed;

  2. (ii)

    \(A^*\) is maximal dissipative;

  3. (iii)

    \(\sigma (A)\subset \{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )\le 0\}\);

  4. (iv)

    \(\Vert (A-\lambda )^{-1}\Vert \le |\mathfrak {R}(\lambda )|^{-1}\) for \(\mathfrak {R}(\lambda )>0\).

We now recall from Hille–Yoshida’s theorem that a maximally dissipative operator \(A_0\) generates a strongly continuous semigroup \(T_0\) on H, and so does \(A_0+K\) for any bounded perturbation K. Let us now recall the following classical properties of strongly continuous semigroup T(t).

Proposition 3.2

(Growth bound, [20] Cor 2.11, p. 258) Let the growth bound of the semigroup be defined as

$$\begin{aligned} w_0=\inf \{w\in {\mathbb {R}}, \exists M_w\ \ \text{ such } \text{ that }\ \ \forall t\ge 0, \ \ \Vert T(t)\Vert \le M_we^{ wt}\}. \end{aligned}$$

Let \(w_{\mathrm{ess}}\) denote the essential growth bound of the semigroup:

$$\begin{aligned} w_{\mathrm{ess}}=\inf \{w\in {\mathbb {R}}, \exists M_w\ \ \text{ such } \text{ that }\ \ \forall t\ge 0, \ \ \Vert T(t)\Vert _{\mathrm{ess}}\le M_we^{ wt}\} \end{aligned}$$

with

$$\begin{aligned} \Vert T(t)\Vert _{\mathrm{ess}}=\inf _{K\in {\mathcal {K}}(H)} \Vert T(t)-K\Vert _{H\rightarrow H} \end{aligned}$$

and \({\mathcal {K}}(H)\) is the ideal of compact operators on H; and let

$$\begin{aligned} s(A)=\mathop {\mathrm{sup}}\{\mathfrak {R}(\lambda ), \ \ \lambda \in \sigma (A)\}. \end{aligned}$$

Then

$$\begin{aligned} w_0=\max \{w_{\mathrm{ess}},s(A)\} \end{aligned}$$

and

$$\begin{aligned} \forall w>w_{\mathrm{ess}}, \ \ \text{ the } \text{ set }\ \ \Lambda _w(A):=\sigma (A)\cap \{\mathfrak {R}(\lambda )>w\}\ \ \text{ is } \text{ finite }. \end{aligned}$$
(3.2)

Moreover, each eigenvalue \(\lambda \in \Lambda _w(A)\) has finite algebraic multiplicity \(m^a_\lambda \): \(\exists k_\lambda \in {\mathbb {Z}}\) such that

$$\begin{aligned}&ker (A-\lambda I)^{k_\lambda }\ne \emptyset ,\qquad \forall k\ge k_\lambda ,\, ker (A-\lambda I)^{k}=ker (A-\lambda I)^{k_\lambda },\\&m^a_\lambda :=dim\,ker (A-\lambda I)^{k_\lambda } \end{aligned}$$

We note that the subspaces \(V_w(A)=\cup _{\lambda \in \Lambda _w(A)} ker (A-\lambda I)^{k_\lambda }\) and \(V_w^\perp (A^*)\) are invariant for A. In particular, \(A\left( D(A)\cap V_w^\perp (A^*)\right) \subset V_w^\perp (A^*)\). The invariance \(V_w(A)\) is immediate. To show that \(A\left( D(A)\cap V_w^\perp (A^*)\right) \subset V_w^\perp (A^*)\) we let \(X\in D(A)\cap V_w^\perp (A^*)\), \(Y\in V_w(A^*)\) and consider \( \langle AX,Y\rangle \). Since \(Y\in D(A^*)\) and \(V_w(A^*)\) is invariant for \(A^*\),

$$\begin{aligned} \langle AX,Y\rangle =\langle X,A^*Y\rangle =0. \end{aligned}$$

We claim the following corollary.

Lemma 3.3

(Perturbative exponential decay) Let \(T_0\) be the strongly continuous semigroup generated by a maximal dissipative operator \(A_0\), and T be the strongly continuous semi group generated by \(A=A_0+K\) where K is a compact operator on H. Then for any \(\delta >0\), the following holds:

  1. (i)

    the set \(\Lambda _\delta (A)=\sigma (A)\cap \{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )> \delta \}\) is finite, each eigenvalue \(\lambda \in \Lambda _\delta (A)\) has finite algebraic multiplicity \(k_\lambda \). In particular, the subspace \(V_\delta (A)\) is finite dimensional;

  2. (ii)

    We have \(\Lambda _\delta (A)=\overline{ \Lambda _\delta (A^*)}\) and \(dim V_\delta (A^*)=dim V_\delta (A)\). The direct sum decomposition

    $$\begin{aligned} H=V_\delta (A)\bigoplus V^\perp _\delta (A^*) \end{aligned}$$
    (3.3)

    is preserved by T(t) and there holds:

    $$\begin{aligned} \forall X\in V^\perp _\delta (A^*), \ \ \Vert T(t)X\Vert \le M_\delta e^{\delta t}\Vert X\Vert . \end{aligned}$$
    (3.4)
  3. (iii)

    The restriction of A to \(V_\delta (A)\) is given by a direct sum of \((m_\lambda \times m_\lambda )_{\lambda \in \Lambda _\delta (A)}\) matrices each of which is the Jordan block associated to the eigenvalue \(\lambda \) and the number of Jordan blocks corresponding to \(\lambda \) is equal to the geometric multiplicity of \(\lambda \)\(m^g_\lambda =dim ker (A-\lambda I)\). In particular, \(m^a_\lambda \le m^g_\lambda k_\lambda \). Each block corresponds to an invariant subspace \(J_\lambda \) and the semigroup T restricted to \(J_\lambda \) is given by the matrix

    $$\begin{aligned} T(t)|_{J_\lambda }=\begin{pmatrix} e^{\lambda t} &{} te^{\lambda t}&{}\cdots &{}t^{m_\lambda -1} e^{\lambda t}\\ 0&{} e^{\lambda t}&{}\cdots &{}t^{m_\lambda -2} e^{\lambda t}\\ \cdots \\ 0&{}0&{}\cdots &{} e^{\lambda t} \end{pmatrix}. \end{aligned}$$

Proof

This is a simple consequence of Proposition 3.2.

Step 1 Perturbative bound. First, since \(A_0\) is maximally dissipative,

$$\begin{aligned} \forall t\ge 0, \ \ \Vert T_0(t)\Vert \lesssim 1 \end{aligned}$$

implies \(w_0(A_0)\le 0\). By Proposition 3.2, \(s(A_0)\le 0\) and

$$\begin{aligned} w_{ess}(T_0)\le 0. \end{aligned}$$

On the other hand, from [20] Prop 2.12 p. 258, compactness of K implies

$$\begin{aligned} w_{\mathrm{ess}}(T)=w_{\mathrm{ess}}(T_0)\le 0. \end{aligned}$$

Let now \(\lambda \in \sigma (A)\) with \(\mathfrak {R}(\lambda )>0\), then the formula

$$\begin{aligned} A-\lambda =A_0+K-\lambda =(A_0-\lambda )(\mathrm{Id}+(A_0-\lambda )^{-1}K) \end{aligned}$$

and invertibility of \((A_0-\lambda )\) imply that \(\lambda \) belongs to the spectrum of the Fredholm operator \(\mathrm{Id} +(A_0-\lambda )^{-1}K\). Therefore, \(\lambda \) is an eigenvalue of A. On the other hand, \(\mathfrak {R}(\lambda )>\delta \) implies \(\mathfrak {R}(\lambda )>\delta >0\ge w_{\mathrm{ess}}(T)\), and hence, by (3.2), there are finitely many eigenvalues with \(\mathfrak {R}(\lambda )>\delta \). In fact, Proposition 3.2 also directly shows that each some \(\lambda \) is an eigenvalue and implies the rest of (i).

Since \(A^*=A_0^*+K^*\) and \(A_0^*\) is maximally dissipative from Lemma 3.1, we can run the same argument as above for \(A^*\). Moreover, \(\sigma (A)=\overline{\sigma (A^*)}\) ([56], Prop. 2.7), (i) is proved.

The argument above, in fact, shows that \(\{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )> \delta \}\cap \{\lambda \in \sigma (A)\}\) is finite, since for every \(\mathfrak {R}(\lambda )>0\) and \(\lambda \in \sigma (A)\), \(\lambda \) is an eigenvalue of A.

Step 2 The first statement of (ii) is standard. We already explained that the subspaces \(V_\delta (A)\) and \(D(A)\cap V_\delta ^\perp (A^*)\) are invariant for A. To prove the direct decomposition we recall that the subspace \(V_\delta (A)\) is the image of H under the spectral projection \(P_\delta (A)\) associated to the set \(\Lambda _\delta (A)\):

$$\begin{aligned} P_\delta (A)=\frac{1}{2\pi i} \int _\Gamma \frac{d\lambda }{\lambda I -A}, \end{aligned}$$

where \(\Gamma \) is an arbitrary contour containing the set \(\Lambda _\delta (A)\). There is a direct decomposition

$$\begin{aligned} H=Im P_\delta (A)\bigoplus ker P_\delta (A). \end{aligned}$$

On the other hand, the adjoint

$$\begin{aligned} P^*_\delta (A)=\frac{1}{2\pi i} \int _{{{\overline{\Gamma }}}} \frac{d\lambda }{\lambda I -A^*}=P_\delta (A^*) \end{aligned}$$

is the spectral projection of \(A^*\) associated to the set \({\overline{\Lambda _\delta (A)}}\). The result is now immediate.

Step 3 Semigroups generated by restriction and conclusion. Let \(V=V_\delta (A)\), \(U=V^\perp _\delta (A^*)\) and P denote the projection on \(V_\delta ^\perp (A^*)\) in the direct decomposition (3.3). Let \({\tilde{A}}\) denote the restriction of A to U with the domain \(D({\tilde{A}})=U\cap D(A)\). By invariance

$$\begin{aligned} \forall X\in U\cap D(A), \ \ {\tilde{A}}X=AX. \end{aligned}$$

Let \({\tilde{T}}\) be the semigroup on U generated by \({\tilde{A}}=A\). Then for all \(X\in D(A)\cap U\), \({\tilde{T}}(t)X\in {\mathcal {C}}^1([0,+\infty ),D({\tilde{A}}))\) is the unique strong solution to the ode

$$\begin{aligned} \frac{dX(t)}{dt}={A}X(t), \ \ X(0)=X. \end{aligned}$$

This implies that \({\tilde{T}}(t)X=T(t)X\) for all \(X\in D(A)\cap U\) and thus for all \(X\in U\) by continuity of the semigroup. By Proposition 3.2 the growth bound of \({\tilde{T}}\) satisfies

$$\begin{aligned} w_0({\tilde{T}})\le \max \{w_{\mathrm{ess}}({{\tilde{T}}}),s({\tilde{A}})\}. \end{aligned}$$

We first argue that

$$\begin{aligned} w_{\mathrm{ess}}({{\tilde{T}}})\le 0. \end{aligned}$$

To prove that we note that we already established that \(w_{\mathrm{ess}}(T)\le 0\). We then fix \(\varepsilon >0\) and, for any \(t\ge 0\) choose a compact operator \(K(t)\in {\mathcal {K}}(H)\) on H such that

$$\begin{aligned} \mathrm{log}\Vert T(t)-K(t)\Vert _{H\rightarrow H}<\varepsilon t + \mathrm{log}M \end{aligned}$$

for some constant M which may depend on \(\varepsilon \). The restriction \({{\tilde{K}}}(t)=PK(t)\) of K(t) to U is a compact operator on U. Then, for any \(t\ge 0\)

$$\begin{aligned} \mathrm{log}\Vert {{\tilde{T}}}(t)-{{\tilde{K}}}(t)\Vert _{U\rightarrow U}= & {} \mathrm{log}\Vert P(T(t)-K(t))\Vert _{U\rightarrow U}\\\le & {} \mathrm{log}\{C _P\Vert T(t)-K(t)\Vert _{H\rightarrow H}\}\\< & {} \mathrm{log}{C _P}+ \mathrm{log}M+ \varepsilon t, \end{aligned}$$

where \(C_P\) denotes the norm of the projector P. The desired conclusion follows.

To show that \(s({{\tilde{A}}})\le \delta \) we assume that \(\lambda \in \sigma ({\tilde{A}})\) with \(\mathfrak {R}(\lambda )>\delta \), then \(\lambda \) is an eigenvalue of \({\tilde{A}}\) and, by invariance of U, \({\lambda }\) is an eigenvalue of A with a non-trivial eigenvector \(\psi \in U\). However, by construction, all such \(\psi \) belong to the subspace \(V=V_\delta (A)\), contradiction. Hence \(s({\tilde{A}})\le \delta \) and Proposition 3.2 yields (3.4).

Finally, part (iii) is completely standard. \(\square \)

We will use Lemma 3.3 in the following form.

Lemma 3.4

(Exponential decay modulo finitely many instabilities) Let \(\delta >0\) and let \(T_0\) be the strongly continuous semigroup generated by a maximal dissipative operator \(A_0\), and T be the strongly continuous semigroup generated by \(A=A_0-\delta +K\) where K is a compact operator on H. Let the (possibly empty) finite set

$$\begin{aligned} \Lambda =\{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )\ge 0\} \cap \{\lambda \ \ \text{ is } \text{ an } \text{ eigenvalue } \text{ of }\ \ A\}=(\lambda _i)_{1\le i\le N} \end{aligned}$$

and let

$$\begin{aligned} H=U\bigoplus V, \end{aligned}$$

where U and V are invariant subspaces for A and V is the image of the spectral projection of A associated to the set \(\Lambda \). Then there exist \(C,\delta _g>0\) such that

$$\begin{aligned} \forall X\in U, \ \ \Vert T(t)X\Vert \le C e^{-\frac{\delta _g}{2} t}\Vert X\Vert . \end{aligned}$$
(3.5)

Proof

We apply Lemma 3.3 to \({\tilde{A}}=A+\delta =A_0+K\) with generates the semi group \({\tilde{T}}\). Hence the set

$$\begin{aligned} \Lambda _{\frac{\delta }{4}}({\tilde{A}})=\left\{ \lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )> \frac{\delta }{4}\right\} \cap \{\lambda \ \ \text{ is } \text{ an } \text{ eigenvalue } \text{ of }\ \ {\tilde{A}}\} \end{aligned}$$

is finite. Moreover

$$\begin{aligned} AX=\lambda X\Leftrightarrow {\tilde{A}}X=(\lambda +\delta )X \end{aligned}$$

and hence

$$\begin{aligned} \Lambda \subset \Lambda _{\frac{\delta }{4}}. \end{aligned}$$

Let

$$\begin{aligned} H=U_\delta \bigoplus V_\delta \end{aligned}$$

be the invariant decomposition of \({{\tilde{A}}}\) (and of A) associated to the set \(\Lambda _{\frac{\delta }{4}}\). Clearly, \(U_\delta \subset U\) and

$$\begin{aligned} U=U_\delta \bigoplus O_\delta , \end{aligned}$$

where \(O_\delta \) is the image of the spectral projection of A associated with the set \(\Lambda _{\frac{\delta }{4}}\setminus \Lambda \). By Lemma 3.3,

$$\begin{aligned} \forall X\in U_\delta , \ \ \Vert {\tilde{T}}(t)X\Vert \le M_\delta e^{\frac{\delta }{4} t}\Vert X\Vert , \end{aligned}$$

which implies

$$\begin{aligned} \forall X\in U_\delta , \ \ \Vert T(t)X\Vert =e^{-\delta t} \Vert {\tilde{T}}(t)X\Vert \le M_\delta e^{-\frac{3\delta }{4} t}\Vert X\Vert . \end{aligned}$$
(3.6)

Let now \(X\in U\). Since \(U_\delta \) is invariant by T and (3.6) yields exponential decay on \(U_\delta \), we assume \(X\in O_\delta \). \(O_\delta \) is an invariant subspace of A generated by the eigenvalues \(\lambda \) with the property that \(-\frac{3}{4}\delta \le \mathfrak {R}(\lambda ) <0\). Let \(\delta _g>0\) be defined as

$$\begin{aligned} -{\delta _g}:=\mathop {\mathrm{sup}}\left\{ \mathfrak {R}(\lambda ): -\frac{3}{4}\delta \le \mathfrak {R}(\lambda ) <0\right\} \end{aligned}$$

From part (iii) of Lemma 3.3,

$$\begin{aligned} \Vert T(t)X\Vert _{O_\delta }\le C \mathop {\mathrm{sup}}_{\mathfrak {R}(\lambda ) < 0} e^{\lambda t} t^{m_\lambda -1}\Vert X\Vert \le C^{-\frac{\delta _g}{2}t}\Vert X\Vert . \end{aligned}$$

This concludes the proof of Lemma 3.4. \(\square \)

Our final result in this section is to set up a Brouwer type argument for the evolution of unstable modes.

Lemma 3.5

Let \(A,\delta _g\) as in Lemma 3.4 with the decomposition

$$\begin{aligned} H=U\bigoplus V \end{aligned}$$

into stable and unstable subspaces Fix a sufficiently large \(t_0>0\) (dependent on A). Let F(tx) such that, \(\forall t\ge t_0\), \(F(t, x)\in V\), depends continuously on x and

$$\begin{aligned} \Vert F(t,x)\Vert \le e^{-\frac{2\delta _g}{3} t} \end{aligned}$$

be given. Let x(t) denote the solution to the ode

$$\begin{aligned} \left| \begin{array}{l} \frac{dx}{dt} = A {x} + F(t,x),\\ x(t_0)=x_0\in V. \end{array}\right. \end{aligned}$$

Then, for any \(x_0\) in the ball

$$\begin{aligned} \Vert x_0\Vert \le e^{-\frac{3\delta _g}{5}t_0}, \end{aligned}$$

we have

$$\begin{aligned} \Vert x(t)\Vert \le e^{-\frac{\delta _g}{2} t},\quad t_0\le t\le t_0+\Gamma \end{aligned}$$
(3.7)

for some large constant \(\Gamma \) (which only depends on A and \(t_0\).) Moreover, there exists \(x^*\in V\) in the same ball as a above such that \(\forall t\ge t_0\) the solution x(t) with initial data \(x(t_0)=x^*\) obeys

$$\begin{aligned} \Vert x(t)\Vert \le e^{-\frac{3\delta _g}{5} t}. \end{aligned}$$

Proof

According to Lemma 3.3 the subspace V can be further decomposed into invariant subspaces on which A is represented by Jordan blocks. We may therefore assume that V is irreducible and corresponds to a Jordan block of A of length \(m_\lambda \) associated with an eigenvalue \(\lambda \) with \({\mathfrak {R}(\lambda )\ge 0}\) and restrict A to V. We decompose A as

$$\begin{aligned} A=\lambda I + N, \end{aligned}$$

where N has the property that \(N^{m_\lambda -1}=0\), and

$$\begin{aligned} e^{tN}=\begin{pmatrix} 1 &{} t&{}\cdots &{}t^{m_\lambda -1} \\ 0&{} 1&{}\cdots &{}t^{m_\lambda -2} \\ \cdots \\ 0&{}0&{}\cdots &{} 1 \end{pmatrix}. \end{aligned}$$

The claim (3.7) follows from the growth on the Jordan block:

$$\begin{aligned} \Vert x(t)\Vert= & {} \left\| e^{(t-t_0)A}x_0+\int _{t_0}^te^{(t-\tau )A}F(\tau ,x)d\tau \right\| \\\le & {} C\Gamma ^{m_\lambda -1}e^{\mathfrak {R}(\lambda )\Gamma }e^{-\frac{3\delta _g t_0}{5}}+\int _{t_0}^{t}C|\tau -t_0| ^{m_\lambda -1}e^{\mathfrak {R}(\lambda )(t-\tau )}e^{-\frac{2}{3}\delta _g\tau }d\tau \\\le & {} C\Gamma ^{m_\lambda -1}e^{\mathfrak {R}(\lambda )\Gamma }e^{-\frac{3\delta _g t_0}{5}} \end{aligned}$$

and hence the size of constant \(\Gamma \) is determined from the inequality

$$\begin{aligned} C\Gamma ^{m_\lambda -1}e^{\mathfrak {R}(\lambda )\Gamma }e^{-\frac{3\delta _g t_0}{5}}\le e^{-\frac{\delta }{2}(t_0+\Gamma )}, \end{aligned}$$

a sufficient condition being

$$\begin{aligned} \Gamma \le \frac{t_0}{2}\left[ \frac{\delta _g}{10\mathfrak {R}(\lambda )+5\delta _g} \right] , \end{aligned}$$

which can be made arbitrarily large by a choice of \(t_0\).

We now define a new variable

$$\begin{aligned} Y(t)=e^{-tN}e^{\frac{19\delta _g}{30} t}x(t). \end{aligned}$$

Since N and A commute,

$$\begin{aligned} \frac{dY}{dt} = \left( \lambda +\frac{19\delta _g}{30}\right) Y + {{\tilde{F}}}(t,Y), \quad Y(t_0)=y, \end{aligned}$$

where \({{\tilde{F}}}(t,Y)=e^{-tN}e^{\frac{19\delta _g}{30} t}F(t,x)\) and

$$\begin{aligned} \Vert {{\tilde{F}}}(t,Y)\Vert \lesssim e^{-\frac{\delta _g}{31} t}. \end{aligned}$$

Since \(t_0\) was chosen to be sufficiently large, we can assume that \(\forall t\ge t_0\)

$$\begin{aligned} \Vert {{\tilde{F}}}(t,Y)\Vert \lesssim \epsilon e^{-\frac{\delta _g}{60} t} \end{aligned}$$

and \(\epsilon < \mathfrak {R}(\lambda )+\frac{19\delta _g}{60}\). We now run a standard Brouwer type argument for Y. For any y such that \(\Vert y\Vert \le 1\) we define the exit time \(t^*\) to be the first time such that \(\Vert Y(t^*)\Vert =1\). If for some y, \(t^*=\infty \), we are done. Otherwise, assume that for all \(\Vert y\Vert \le 1\), \(t^*<\infty \) and define the map \(\Phi : B\rightarrow S\) as \(\Phi (y)=Y(t^*)\) mapping the unit ball to the unit sphere. Note that \(\Phi \) is the identity map on the boundary of B. To prove continuity of \(\Phi \) we compute

$$\begin{aligned} \frac{d\Vert Y\Vert ^2}{dt}(t^*)=2\mathfrak {R}(\lambda ) +\frac{19\delta _g}{15}+ 2 \mathfrak {R}\langle {{\tilde{F}}}(t^*,Y(t^*)),Y(t^*)\rangle \ge \frac{19\delta _g}{30}>0. \end{aligned}$$

This is the outgoing condition which implies continuity. The Brouwer argument applies and shows that such \(\Phi \) can not exist. We now reinterpret the result in terms of x. We have shown existence of the data \(x^*\) such that the corresponding solution x(t) has the property that \(\forall t\ge t_0\),

$$\begin{aligned} \Vert e^{-tN} x(t)\Vert \le e^{-\frac{19\delta _g}{30} t}. \end{aligned}$$

Now \(e^{-tN}\) is an invertible operator with the inverse given by \(e^{tN}\) and its norm bounded by \(C t^{m_\lambda -1}\). The result follows immediately. We note that the resulting solution x(t) has initial data \(x(t_0)\) in the ball \( \Vert x(t_0)\Vert \le e^{-\frac{3\delta _g}{5}t_0}. \) \(\square \)

3.2 Linearized equations

Recall the exact linearized flow (2.25) which we rewrite:

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho =H_1\rho -H_2\Lambda \rho -\rho _P\Delta \Psi -2\nabla \rho _P\cdot \nabla \Psi -\rho \Delta \Psi -2\nabla \rho \cdot \nabla \Psi \\ \partial _\tau \Psi =b^2\frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\left\{ H_2\Lambda \Psi +\mu (r-2)\Psi +(p-1)\rho _P^{p-2}\rho \right. \\ \qquad \qquad \left. +|\nabla \Psi |^2+\text {NL}(\rho )\right\} . \end{array}\right. \end{aligned}$$

Our aim for the remainder of the section is to find a Hilbert space in which the linearized operator is accretive modulo a compact perturbation in order to apply the general results of the previous section.

We introduce the new unknown

$$\begin{aligned} \Phi =\rho _{P}\Psi \end{aligned}$$
(3.8)

and obtain equivalently using (2.24):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho =H_1\rho -H_2\Lambda \rho -\Delta \Phi +H_3\Phi +G_\rho ,\\ \partial _\tau \Phi =-(p-1)Q\rho -H_2\Lambda \Phi +(H_1-\mu (r-2))\Phi +G_\Phi \end{array}\right. \end{aligned}$$
(3.9)

with

$$\begin{aligned} Q=\rho _{P}^{p-1},\quad H_3=\frac{\Delta \rho _{P}}{\rho _{P}} \end{aligned}$$
(3.10)

and the nonlinear terms:

$$\begin{aligned} \left| \begin{array}{ll} G_\rho =-\rho \Delta \Psi -2\nabla \rho \cdot \nabla \Psi ,\\ G_\Phi =-\rho _P(|\nabla \Psi |^2+\text {NL}(\rho ))+\frac{b^2\rho _P}{\rho _{\mathrm{Tot}}}\Delta \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$
(3.11)

We transform (3.9) into a wave equation for \(\Phi \) and compute:

$$\begin{aligned} \partial _\tau ^2\Phi= & {} -(p-1)Q(H_1\rho -H_2\Lambda \rho -\Delta \Phi +H_3\Phi +G_\rho )+\partial _\tau G_\Phi \\&-H_2\Lambda \partial _\tau \Phi + (H_1-\mu (r-2))\partial _\tau \Phi \\= & {} -(p-1)Q(-\Delta \Phi +H_3\Phi ) -H_2\Lambda \partial _\tau \Phi + (H_1-\mu (r-2))\partial _\tau \Phi \\&+(p-1)Q({-H_1+}H_2\Lambda )\left\{ \frac{1}{(p-1)Q}\left[ -\partial _\tau \Phi -H_2\Lambda \Phi \right. \right. \\&\left. \left. +(H_1-\mu (r-2))\Phi +G_\Phi \right] \right\} +\partial _\tau G_\Phi -(p-1)QG_\rho \\= & {} (p-1)Q\Delta \Phi -H_2^2\Lambda ^2\Phi -2H_2\Lambda \partial _\tau \Phi + A_1\Lambda \Phi +A_2\partial _\tau \Phi +A_3 \Phi \\&+\partial _\tau G_\Phi -\left( H_1+H_2\frac{\Lambda Q}{Q}\right) G_\Phi +H_2\Lambda G_\Phi -(p-1)QG_\rho \end{aligned}$$

with

$$\begin{aligned}\left| \begin{array}{llll} A_1=H_2H_1-H_2\Lambda H_2+H_2(H_1-\mu (r-2))+H_2^2\frac{\Lambda Q}{Q},\\ A_2=2H_1-\mu (r-2)+H_2\frac{\Lambda Q}{Q},\\ A_3=-(H_1-{\mathscr {e}})H_1+H_2\Lambda H_1-H_2(H_1-\mu (r-2))\frac{\Lambda Q}{Q} - (p-1)QH_3. \end{array}\right. \end{aligned}$$

In this section we focus on deriving decay estimates for (3.9).

Remark 3.6

(Null coordinates and red shift) We note that the principal symbol of the above wave equation is given by the second order operator

$$\begin{aligned} \Box _Q:=\partial _\tau ^2 - ((p-1)Q-H_2^2 Z^2)\partial _Z^2+2H_2Z \partial _Z\partial _\tau . \end{aligned}$$

In the variables of Emden transform this can be written equivalently as

$$\begin{aligned} \Box _Q=\partial _\tau ^2 - \mu ^2\left[ \sigma ^2-(1-w)^2\right] \partial _{{\varkappa }}^2+2\mu (1-w)\partial _{ {\varkappa }}\partial _\tau . \end{aligned}$$

The two principal null direction associated with the above equation are

$$\begin{aligned} L=\partial _\tau +\mu \left[ (1-w)-\sigma \right] \partial _{{\varkappa }},\qquad {\underline{L}}=\partial _\tau +\mu \left[ (1-w)+\sigma \right] \partial _{{\varkappa }} \end{aligned}$$

so that

$$\begin{aligned} \Box _Q=L{\underline{L}}. \end{aligned}$$

We observe that at P2, we have \(L=\partial _\tau \) and the surface \(Z=Z_2\) is a null line (cone, if we view from the point of view of the higher dimensional space where a point \((\tau ,Z)\) is in fact a \((d-1)\)-dimensional sphere). Moreover, the associated acoustical metric is

$$\begin{aligned} g_Q=\mu ^2\Delta d\tau ^2 -2\mu (1-w) d\tau d{\varkappa } +d{{\varkappa }}^2,\qquad \Delta =(1-w)^2-\sigma ^2 \end{aligned}$$

for which \(\partial _\tau \) is a Killing field (generator of translation symmetry). Therefore, \(Z=Z_2\) is a Killing horizon (generated by a null Killing field.) We can make it even more precise by transforming the metric \(g_Q\) into a slightly different form by defining the coordinate s:

$$\begin{aligned} s=\mu \tau -f({\varkappa }),\quad f'=\frac{1-w}{\Delta } \end{aligned}$$

so that

$$\begin{aligned} g_Q=\Delta (ds)^2-\frac{\sigma ^2}{\Delta } d{\varkappa }^2 \end{aligned}$$

and then the coordinate \(x^*\):

$$\begin{aligned} x^*=\int \frac{\sigma }{\Delta } d{\varkappa } \end{aligned}$$

so that

$$\begin{aligned} g_Q=\Delta \, d(s+x^*)\, d(s-x^*) \end{aligned}$$

and \(s+x^*\) and \(s-x^*\) are the null coordinates of \(g_Q\). The Killing horizon \(Z=Z_2\) corresponds to \(x^*=-\infty \) and \(\Delta \sim e^{Cx^*}\) for some positive constant C. In this form, near \(Z_2\) the metric \(g_Q\) resembles the \(1+1\)-quotient Schwarzschild metric near the black hole horizon. Note that the region \(Z>Z_2\) corresponds to the interior of a black hole in a sense that the null geodesics of the acoustical metric never leave that region.

The associated surface gravity \(\kappa \) which can be computed according to

$$\begin{aligned} \kappa&=\frac{\partial _{x^*} \Delta }{2\Delta }|_{P2} =\frac{\partial _{x} \Delta }{2\sigma }|_{P2}= \frac{-w'(1-w)-\sigma '\sigma }{\sigma }|_{P2}\\&=(-w'-\sigma ')|_{P2}=1-w-\Lambda w -\frac{(1-w)F}{\sigma }|_{P2}>0. \end{aligned}$$

This is precisely the positivity condition (2.21) (at P2). The positivity of surface gravity implies the presence of the red shift effect along \(Z=Z_2\) both as an optical phenomenon for the acoustical metric \(g_Q\) and also as an indicator of local monotonicity estimates for solutions of the wave equation \(\Box _Q \varphi =0\), [17]. Near \(Z_2\), the null characteristics spread out and the monotonicity estimates can be captured with the energy estimates based on a multiplier transversal to the set \(Z=Z_2\), while the standard energy estimates based on the multiplier \(\partial _\tau \) would be degenerate. The complication in the analysis below is the presence of lower order terms in the wave equation as well as the need for global in space estimates.

3.3 The linearized operator

The degeneracy of wave operator \(\Box _Q\) is a feature of the chosen coordinate system and, specifically, of the fact that \(\partial _\tau \) is tangent to the set \(Z=Z_2\). We can remedy this by adding to \(\partial _\tau \) a small amount of \(\Lambda \)-derivative near \(Z_2\). The precise technical implementation is as follows.

Pick a small enough parameter

$$\begin{aligned} 0<a\ll 1 \end{aligned}$$

and consider the new variable

$$\begin{aligned} T=\partial _\tau \Phi +aH_2\Lambda \Phi , \end{aligned}$$
(3.12)

then

$$\begin{aligned} \partial _\tau T= & {} \partial ^2_\tau \Phi +aH_2\Lambda \partial _\tau \Phi = \partial ^2_\tau \Phi +aH_2\Lambda (T-aH_2\Lambda \Phi )\\= & {} \partial ^2_\tau \Phi +aH_2\Lambda T-a^2H_2\Lambda H_2\Lambda \Phi -a^2H_2^2\Lambda ^2\Phi , \end{aligned}$$

which yields the \((T,\Phi )\) equation

$$\begin{aligned} \partial _\tau \Phi =T-aH_2\Lambda \Phi \end{aligned}$$

and

$$\begin{aligned} \partial _\tau T= & {} (p-1)Q\Delta \Phi -H_2^2\Lambda ^2\Phi -2H_2\Lambda (T-aH_2\Lambda \Phi ) \\&+ A_1\Lambda \Phi +A_2(T-aH_2\Lambda \Phi )+A_3 \Phi \\&+ aH_2\Lambda T-a^2H_2\Lambda H_2\Lambda \Phi -a^2H_2^2\Lambda ^2\Phi + G_T\\= & {} (p-1)Q\Delta \Phi -(1-a)^2H_2^2\Lambda ^2\Phi \\&+\tilde{A_2}\Lambda \Phi +A_3\Phi -(2- a)H_2\Lambda T +A_2T+ G_T \end{aligned}$$

with

$$\begin{aligned} G_T=\partial _\tau G_\Phi -\left( H_1+H_2\frac{\Lambda Q}{Q}\right) G_\Phi +H_2\Lambda G_\Phi -(p-1)QG_\rho \end{aligned}$$
(3.13)

and

$$\begin{aligned} {\tilde{A}}_2=A_1+(2a-a^2)H_2\Lambda H_2-a A_2H_2. \end{aligned}$$

We rewrite these equations in vectorial form

$$\begin{aligned} \partial _\tau X={\mathcal {M}} X +G, \ \ X=\left| \begin{array}{ll}\Phi \\ T\end{array}\right. , \ \ G=\left| \begin{array}{ll}0\\ G_T\end{array}\right. \end{aligned}$$
(3.14)

with

$$\begin{aligned} {\mathcal {M}}= \left( \begin{array}{cc} -aH_2\Lambda &{} 1 \\ (p-1)Q\Delta - (1-a)^2H_2^2\Lambda ^2+\tilde{A_2}\Lambda +A_3 &{}\ -(2-a)H_2\Lambda +A_2\end{array}\right) .\nonumber \\ \end{aligned}$$
(3.15)

3.4 Shifted measure

The fine structure of the operator (3.15) involves the understanding of the associated light cone.

Lemma 3.7

(Shifted measure) LetFootnote 5

$$\begin{aligned} D_a=(1-a)^2(w-1)^2-\sigma ^2 \end{aligned}$$
(3.16)

then for \(0<a<a^*\) small enough, there exists a \({\mathcal {C}}^1\) map \(a\mapsto Z_a\) with

$$\begin{aligned} Z_{a=0}=Z_2, \ \ \frac{\partial Z_a}{\partial a}>0 \end{aligned}$$

such that

$$\begin{aligned} \left| \begin{array}{l} D_a(Z_a)=0,\\ -D_a(Z)>0\ \ \text{ on }\ \ 0\le Z<Z_a,\\ \mathop {\mathrm{lim}}_{Z\rightarrow 0}Z^2(-D_a)>0. \end{array}\right. \end{aligned}$$
(3.17)

Proof of Lemma 3.7

We recall the notations of the Emden transform:

$$\begin{aligned} \left| \begin{array}{l} x=\mathrm{log}Z,\\ \mu =\frac{1-{{\mathscr {e}}}}{2},\\ F=\sigma +\Lambda \sigma ,\\ (p-1)Q=\mu ^2Z^2\sigma ^2, \ \ \frac{\Lambda Q}{Q}=2+2\frac{\Lambda \sigma }{\sigma }=\frac{2{(\sigma +\Lambda \sigma )}}{\sigma },\\ (p-1)\partial _ZQ=(p-1)\frac{\Lambda Q}{Q}\frac{Q}{Z}=2\mu ^2Z\sigma ^2\left( 1+\frac{\Lambda \sigma }{\sigma }\right) \\ \quad \qquad \qquad \qquad =2\mu ^2Z\sigma {(\sigma +\Lambda \sigma )},\\ H_2=\frac{1-{{\mathscr {e}}}}{2}+2\frac{\partial _Z\Psi _P}{Z}=\mu (1-w),\\ H_1=H_2\frac{\Lambda \rho _P}{\rho _P}=\frac{H_2}{p-1}\frac{\Lambda Q}{Q}=\frac{2\mu {(\sigma +\Lambda \sigma )}(1-w)}{(p-1)\sigma },\\ D=(w-1)^2-\sigma ^2. \end{array}\right. \nonumber \\ \end{aligned}$$
(3.18)

Step 1 Values of derivatives at P2. Let

$$\begin{aligned} \Delta =(w-1)^2-\sigma ^2. \end{aligned}$$

Let the variables

$$\begin{aligned} w=w_2+W, \ \ \sigma =\sigma _2+\Sigma , \end{aligned}$$

then near P2:

$$\begin{aligned} W=c_-\Sigma +O(\Sigma ^2). \end{aligned}$$

Let

$$\begin{aligned} \left| \begin{array}{llll} c_1=\partial _W\Delta _1(P_2),\\ c_2=\partial _W\Delta _2(P_2),\\ c_3=\partial _\Sigma \Delta _1(P_2),\\ c_4=\partial _\Sigma \Delta _2(P_2)=-2\sigma _2^2. \end{array}\right. \end{aligned}$$
(3.19)

Then, in our range of parameters,

$$\begin{aligned} c_1<0, \ \ c_2<0, \ \ c_3<0,\ \ c_4<0, \end{aligned}$$
(3.20)

and we have

$$\begin{aligned} \left| \begin{array}{ll} c_2c_-+c_4=\lambda _+,\\ c_2c_++c_4=\lambda _-,\\ c_{\pm }=\frac{c_1c_\pm +c_3}{c_2c_\pm +c_4}, \end{array}\right. \end{aligned}$$
(3.21)

which imply

$$\begin{aligned} c_1c_-+c_3=c_-(c_2c_-+c_4)=c_-\lambda _+ \end{aligned}$$

as well as

$$\begin{aligned} -1<c_-<0<c_+, \ \ \lambda _-<\lambda _+<0, \end{aligned}$$
(3.22)

see Lemmas 2.8 and 2.9 in [43].

We compute

$$\begin{aligned} \left| \begin{array}{l} \Delta _1=c_1W+c_3\Sigma +O(W^2+\Sigma ^2)=(c_1c_-+c_3)\Sigma +O(\Sigma ^2),\\ \Delta _2=c_2W+c_4\Sigma +O(W^2+\Sigma ^2)=(c_2c_-+c_4)\Sigma +O(\Sigma ^2),\\ \Delta =(1-w_2-W)^2-(\sigma _2+\Sigma )^2=(\sigma _2-W)^2-(\sigma _2+\Sigma )^2\\ \,\,\quad =-2\sigma _2(c_-+1)\Sigma +O(\Sigma ^2) \end{array}\right. \end{aligned}$$

This yields

$$\begin{aligned} \left| \begin{array}{l} \frac{dw}{dx}=-\frac{\Delta _1}{\Delta }=-\frac{c_1c_-+c_3+O(\Sigma )}{-2\sigma _2(1+c_-)+O(\Sigma )}=\frac{|c_-||\lambda _+|}{2\sigma _2(1+c_-)}+O(\Sigma ),\\ \frac{d\sigma }{dx}=-\frac{\Delta _2}{\Delta }=-\frac{c_2c_-+c_4+O(\Sigma )}{-2\sigma _2(1+c_-)+O(\Sigma )}=-\frac{|\lambda _+|}{2\sigma _2(1+c_-)}+O(\Sigma ), \end{array}\right. \end{aligned}$$
(3.23)

and hence

$$\begin{aligned} Z_2\frac{d\Delta }{dZ}(Z_2)= & {} \frac{d\Delta }{dx}(P_2)=-2(1-w_2)\frac{dw}{dx}(P_2)-2\sigma _2\frac{d\sigma }{dx}(P_2) \nonumber \\= & {} -2\sigma _2\frac{|c_-||\lambda _+|}{2\sigma _2(1+c_-)}-2\sigma _2\left[ -\frac{|\lambda _+|}{2\sigma _2(1+c_-)}\right] \nonumber \\= & {} \frac{|\lambda _+|}{1+c_-}(1-|c_-|)\nonumber \\= & {} |\lambda _+|>0 \end{aligned}$$
(3.24)

Step 2 Computation of \(Z_a\). Let \(D_0(Z)=\Delta (Z)\), we have \(D_0'(Z_2)>0\) from (3.24) and hence by the implicit function theorem applied to the function \(F(a, Z)=D_a(Z)\) at \((a, Z)=(0, Z_2)\) where \(D_0(Z_2)=0\), we infer for all a small enough the existence of a locally unique solution \(Z_a\) to

$$\begin{aligned} D_a(Z_a)=0. \end{aligned}$$
(3.25)

Furthermore, \(Z_a\) is \({\mathcal {C}}^1\) in a neighborhood of \(a=0\) and its derivative is given by

$$\begin{aligned} \frac{\partial Z_a}{\partial a}_{|a=0} = -\left( \frac{\frac{\partial D_a(Z)}{\partial a}}{\frac{\partial D_a(Z)}{\partial Z}}\right) _{a=0, \, Z=Z_2}= \frac{2\sigma _2^2}{D_0'(Z_2)}>0. \end{aligned}$$

Thus

$$\begin{aligned} \frac{\partial Z_a}{\partial a}_{|a=0}>0,\ \ Z_a>Z_2\ \ \text{ for }\ \ 0<a\ll 1, \ \ D_a'(Z_a)>0. \end{aligned}$$
(3.26)

We now observe

$$\begin{aligned} D_a(Z)= ((1-a)(1-w)+\sigma )((1-a)(1-w)-\sigma ) \end{aligned}$$

so that \(D_a(Z)\) is of the sign of \((1-a)(1-w)-\sigma \) since \(w<1\) and \(\sigma >0\). Now from (3.23):

$$\begin{aligned} \frac{d}{dx}\Big ((1-a)(1-w)-\sigma \Big )= & {} -(1-a)\frac{|c_-||\lambda _+|}{2\sigma _2(1+c_-)}+\frac{|\lambda _+|}{2\sigma _2(1+c_-)}\\= & {} \frac{|\lambda _+|}{2\sigma _2(1+c_-)}[1-(1-a)|c_-|]>0. \end{aligned}$$

Thus, \((1-a)(1-w)-\sigma \) is increasing on \((0,Z_a]\) and vanishes at \(Z=Z_a\) so that

$$\begin{aligned} D_a(Z)<0\quad \text {on}(0, Z_a). \end{aligned}$$

Moreover, we have in view of the behavior of \(\sigma \) and w as \(Z\rightarrow 0_+\), see Lemma 3.1 in [43],

$$\begin{aligned} \mathop {\mathrm{lim}}_{Z\rightarrow 0_+}Z^2(-D_a(Z)) = \mathop {\mathrm{lim}}_{Z\rightarrow 0_+}Z^2\sigma ^2=1>0. \end{aligned}$$

This concludes the proof of (3.17). \(\square \)

3.5 Commuting with derivatives

We define

$$\begin{aligned} T_k=\Delta ^kT, \ \ \Phi _k=\Delta ^k\Phi . \end{aligned}$$

Lemma 3.8

(Commuting with derivatives) Let \(k\in {\mathbb {N}}\). There exists a smooth measureFootnote 6g defined for \(Z\in [0,Z_a]\) such that the following holds. Let the elliptic operator

$$\begin{aligned} {\mathcal {L}}_g\Phi _k=\frac{\mu ^2}{gZ^{d-1}}\partial _Z\left( Z^{d-1}Z^2g(-D_a) \partial _Z\Phi _k\right) , \end{aligned}$$

then there holds

$$\begin{aligned} \Delta ^k({\mathcal {M}}X)={\mathcal {M}}_k\left| \begin{array}{ll}\Phi _k\\ T_k\end{array}\right. +\widetilde{{\mathcal {M}}}_k X \end{aligned}$$
(3.27)

with

$$\begin{aligned} {\mathcal {M}}_k\left| \begin{array}{ll}\Phi _k\\ T_k\end{array}\right. =\left| \begin{array}{ll}-aH_2\Lambda \Phi _k-2ak(H_2+\Lambda H_2)\Phi _k+ T_k\\ {\mathcal {L}}_g \Phi _k-(2-a)H_2\Lambda T_k-2k(2-a)(H_2+\Lambda H_2) T_k +A_2T_k,\end{array}\right. \end{aligned}$$

where \(\widetilde{{\mathcal {M}}}_k\) satisfies the following pointwise bound

$$\begin{aligned} |\widetilde{{\mathcal {M}}}_kX|\lesssim _k \left| \begin{array}{ll} \displaystyle \sum _{j=0}^{2k-1}|\partial ^{j}_Z\Phi |,\\ \displaystyle \sum _{j=0}^{2k}|\partial ^{j}_Z\Phi |+\sum _{j=0}^{2k-1}|\partial _Z^jT|. \end{array}\right. \end{aligned}$$
(3.28)

Moreover, \(g>0\) in \([0,Z_a)\) and admits the asymptotics:

$$\begin{aligned} \left| \begin{array}{ll}g(Z)=1+O(Z^2)\ \ \text{ as }\ \ Z\rightarrow 0,\\ g(Z)=c_{a,d,r,\ell }(Z_a-Z)^{c_g}\left[ 1+O(Z-Z_a)\right] \ \text{ as }\ \ Z\uparrow Z_a, \end{array}\right. \end{aligned}$$
(3.29)

with \(c_{a,d,r,\ell }>0\) and

$$\begin{aligned} c_g>0 \end{aligned}$$
(3.30)

for all \(k\ge k_1\) large enough and \(0<a<a^*\) small enough. Finally, note that g and \(c_g\) depend on k.

Proof

This is a direct computation.

Step 1 Proof of (3.27), (3.28). We recall (C.1):

$$\begin{aligned}{}[\Delta ^k,V]\Phi -2k\nabla V\cdot \nabla \Delta ^{k-1}\Phi =\sum _{|\alpha |+|\beta |=2k,|\beta |\le 2k-2}c_{k,\alpha ,\beta }\partial ^\alpha V\partial ^\beta \Phi , \end{aligned}$$

which together with the commutator formulas

$$\begin{aligned} \left| \begin{array}{l} [\Delta ^k,\Lambda ]=2k\Delta ^k,\ \ [\partial _Z,\Lambda ]=\partial _Z,\\ \Lambda ^2=Z^2\Delta -(d-2)\Lambda ,\\ \partial _Z\Lambda =\frac{\Lambda ^2}{Z}=Z\Delta -(d-2)\partial _Z \end{array}\right. \end{aligned}$$
(3.31)

yields

$$\begin{aligned} \Delta ^k(V\Lambda \Phi )= & {} V\Delta ^k(\Lambda \Phi )+2k\nabla V\cdot \nabla \Delta ^{k-1}\Lambda \Phi \\&+\sum _{|\alpha |+|\beta |=2k,|\beta |\le 2k-2}c_{k,\alpha ,\beta }\partial ^\alpha V\partial ^\beta \Lambda \Phi . \end{aligned}$$

and

$$\begin{aligned}&2k\nabla V\cdot \nabla \Delta ^{k-1}\Lambda \Phi = 2k\partial _ZV\partial _Z\left[ \Lambda \Delta ^{k-1}\Phi +2(k-1)\Phi _{k-1}\right] \\&\quad = 2k\partial _ZV\left[ (Z\Delta -(d-2)\partial _Z)\Phi _{k-1}+2(k-1)\partial _Z\Phi _{k-1}\right] \\&\quad = 2k\Lambda V\Phi _k+2k(2k-2-d+2)\partial _ZV\partial _Z\Phi _{k-1} \end{aligned}$$

from which for \(0\le Z\le Z_a\):

$$\begin{aligned} \Delta ^k(V\Lambda \Phi )=V(2k+\Lambda ) \Phi _k+2k\Lambda V\Phi _k+O_{k,a}\left( \sum _{j=0}^{2k-1}|\partial ^{j}_Z\Phi |\right) . \end{aligned}$$

We then use

$$\begin{aligned} {[}\Delta ^k,\Lambda ^2]= & {} \Delta ^k\Lambda ^2-\Lambda ^2\Delta ^k=[\Delta ^k,\Lambda ]\Lambda +\Lambda \Delta ^k\Lambda -\Lambda (-[\Delta ^k,\Lambda ]+\Delta ^k\Lambda )\\= & {} 2k\Delta ^k\Lambda +2k\Lambda \Delta ^k=4k^2\Delta ^k+4k\Lambda \Delta ^k \end{aligned}$$

to compute similarly:

$$\begin{aligned} \Delta ^k(V\Lambda ^2 \Phi )= & {} V\Delta ^k(\Lambda ^2 \Phi )+2k\nabla V\cdot \nabla \Delta ^{k-1}\Lambda ^2\Phi \\&+\sum _{|\alpha |+|\beta |=2k,|\beta |\le 2k-2}c_{k,\alpha ,\beta }\partial ^\alpha V\partial ^\beta \Lambda \Phi \\= & {} V\left[ \Lambda ^2\Phi _k+4k^2\Phi _k+4k\Lambda \Phi _k\right] \\&+2k\partial _ZV\partial _Z\Delta ^{k-1}\Lambda ^2\Phi +O\left( \sum _{j=0}^{2k}|\partial ^{j}_Z\Phi |\right) \end{aligned}$$

and

$$\begin{aligned} \partial _Z\Delta ^{k-1}\Lambda ^2\Phi= & {} \partial _Z\left[ \Lambda ^2\Phi _{k-1}+4(k-1)^2\Phi _{k-1}+4(k-1)\Lambda \Phi _{k-1}\right] \\= & {} \partial _Z(Z^2\Phi _k-(d-2)\Lambda \Phi _{k-1})+O\left( \sum _{j=0}^{2k}|\partial ^{j}_Z\Phi |\right) \\= & {} Z\Lambda \Phi _k+O\left( \sum _{j=0}^{2k}|\partial ^{j}_Z\Phi |\right) \end{aligned}$$

and hence

$$\begin{aligned} \Delta ^k(V\Lambda ^2 \Phi )= & {} V\left[ \Lambda ^2\Phi _k+4k^2\Phi _k+4k\Lambda \Phi _k\right] \\&+2k\Lambda V\Lambda \Phi _k+O\left( \sum _{j=0}^{2k}|\partial ^{j}_Z\Phi |\right) . \end{aligned}$$

Recalling the definition of the operator (3.15), we obtain (3.27), (3.28) with

$$\begin{aligned} {\mathcal {M}}_k\left| \begin{array}{ll}\Phi _k\\ T_k\end{array}\right. = \left| \begin{array}{ll}\!\!\! -aH_2\Lambda \Phi _k -2ak(H_2+\Lambda H_2)\Phi _k+ T_k\\ (p-1)Q\Delta \Phi _k-(1-a)^2H_2^2\Lambda ^2\Phi _k+A_k\Lambda \Phi _k -(2-a)H_2\Lambda T_k\\ \quad -2k(2-a)(H_2+\Lambda H_2) T_k +A_2T_k \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} A_k=2k(p-1)\frac{\partial _ZQ}{Z}-(1-a)^24kH_2\left[ H_2+\Lambda H_2\right] +{\tilde{A}}_2. \end{aligned}$$

Step 2 Equation for the measure. We compute using (3.18), (3.16):

$$\begin{aligned}&(p-1)Q\Delta \Phi _k-(1-a)^2H_2^2\Lambda ^2\Phi _k\\&\quad = \mu ^2Z^2\sigma ^2\left( \partial _Z^2\Phi _k+\frac{d-1}{Z}\partial _Z\Phi _k\right) \\&\qquad -\mu ^2(1-w)^2(1-a)^2\left( Z^2\partial _Z^2\Phi _k+\Lambda \Phi _k\right) \\&\quad = \mu ^2\left\{ -Z^2D_a\partial _Z^2\Phi _k+Z\partial _Z\Phi _k\left[ (d-1)\sigma ^2-(1-a)^2(1-w)^2\right] \right\} \end{aligned}$$

and, using \(F=\sigma +\Lambda \sigma \),

$$\begin{aligned} A_k= & {} 2k(p-1)\frac{\partial _ZQ}{Z}-(1-a)^24kH_2\left[ H_2+\Lambda H_2\right] +{\tilde{A}}_2\\= & {} 4k\mu ^2\left[ \sigma F-(1-a)^2(1-w)+(1-a)^2(1-w)(w+\Lambda w)\right] +{\tilde{A}}_2 \end{aligned}$$

and hence:

$$\begin{aligned}&(p-1)Q\Delta \Phi _k-(1-a)^2H_2^2\Lambda ^2\Phi _k+A_k\Lambda \Phi _k\\&\quad = -\mu ^2Z^2D_a\partial _Z^2\Phi _k + \Lambda \Phi _k\Bigg [\mu ^2\left( (d-1)\sigma ^2-(1-a)^2(1-w)^2\right) \\&\qquad +4k\mu ^2\left[ \sigma F-(1-a)^2(1-w)+(1-a)^2(1-w)(w+\Lambda w)\right] +{\tilde{A}}_2\Bigg ]. \end{aligned}$$

We compute the measure

$$\begin{aligned}&\frac{\mu ^2}{gZ^{d-1}}\partial _Z\left( Z^{d-1}Z^2g(-D_a) \Phi _k'\right) \\&\quad =\mu ^2Z^2(-D_a)\partial _Z^2\Phi _k-\mu ^2\Lambda \Phi _k\left( (d+1)D_a+\frac{g'}{g}ZD_a+ZD_a'\right) \end{aligned}$$

and hence the relation:

$$\begin{aligned}&-\mu ^2\left( (d+1)D_a+\frac{g'}{g}ZD_a+ZD_a'\right) \\&\quad = \mu ^2\left( (d-1)\sigma ^2-(1-w)^2\right) +4k\mu ^2\left[ \sigma F-(1-a)^2(1-w)\right. \\&\qquad \left. +(1-a)^2(1-w)(w+\Lambda w)\right] +{\tilde{A}}_2. \end{aligned}$$

Equivalently:

$$\begin{aligned} (-D_a)\frac{\Lambda g}{g}=- {\mathcal {G}} \end{aligned}$$
(3.32)

with

$$\begin{aligned} {\mathcal {G}}= & {} -(d-1)\sigma ^2+(1-w)^2-(d+1)D_a-\Lambda D_a\nonumber \\&+4k\left[ (1-a)^2(1-w) -\sigma F-(1-a)^2(1-w)(w+\Lambda w)\right] \nonumber \\&-\frac{{\tilde{A}}_2}{\mu ^2}. \end{aligned}$$
(3.33)

Step 3 Asymptotics of the measure. We now solve (3.32). Near the origin, the normalization (2.17) and (3.18) yield

$$\begin{aligned} \sigma =\frac{\sqrt{p-1}}{\mu Z}\left[ 1+O(Z^2)\right] , \ \ F=\sigma +\Lambda \sigma =O(Z), \ \ -D_a=\sigma ^2+O(1). \end{aligned}$$

We compute

$$\begin{aligned} \frac{\tilde{A_2}}{\mu ^2}=O\left( \frac{|F|}{\sigma }+|\Lambda w|+|a|\right) =O(1) \end{aligned}$$

and hence

$$\begin{aligned} {\mathcal {G}}= & {} -(d-1)\sigma ^2 -(d+1)(-\sigma ^2)-\Lambda (-\sigma ^2)\\&+O\left( 1+|a|+\sigma |F|+|w|+|\Lambda w|\right) \\= & {} 2\sigma F+O(1) = O(1), \end{aligned}$$

which, recalling (3.17), yields:

$$\begin{aligned} -\frac{{\mathcal {G}}}{(-D_a)}=\frac{O(1)}{\sigma ^2+O(1)}=O(Z^2) \end{aligned}$$

and we may therefore choose explicitly:

$$\begin{aligned} g=e^{\int _0^Z \left[ -\frac{{\mathcal {G}}}{(-D_a)}\right] \frac{d\tau }{\tau }}. \end{aligned}$$

To compute the behavior near \(Z_a\), recall from (3.25) (3.26) that we have

$$\begin{aligned} D_a(Z_a)=0, \ \ D_a'(Z_a)>0. \end{aligned}$$

We infer in the neighborhood of \(Z=Z_a\)

$$\begin{aligned} \frac{\partial _Zg}{g}= & {} \frac{{\mathcal {G}}}{ZD_a}= \frac{{\mathcal {G}}(Z_a)}{\Lambda D_a(Z_a)}\frac{1+O(Z-Z_a)}{Z-Z_a} \nonumber \\= & {} \left( \frac{{\mathcal {G}}(Z_2)}{\Lambda D_0(Z_2)}+O(|a|)\right) \frac{1+O(Z-Z_a)}{Z-Z_a}. \end{aligned}$$
(3.34)

The fundamental computation is then at P2 using (3.23):

$$\begin{aligned}&\left[ (1-w) -\sigma {(\sigma +\Lambda \sigma )}-(1-w)(w+\Lambda w)\right] \\&\quad =(1-w_2)(1-w_2-\Lambda w)-\sigma _2(\sigma _2+\Lambda \sigma )\\&\quad = \sigma _2(\sigma _2-\Lambda w)-\sigma _2(\sigma _2+\Lambda \sigma )=\sigma _2(-\Lambda w-\Lambda \sigma )\\&\quad = \sigma _2\left[ -\frac{|c_-||\lambda _+|}{2\sigma _2(1+c_-)}+\frac{|\lambda _+|}{2\sigma _2(1+c_-)}\right] =\frac{|\lambda _+|}{2}>0. \end{aligned}$$

Hence from (3.33)

$$\begin{aligned} {\mathcal {G}}(Z_2)=2k(|\lambda _+|+O(a))+O(1) \end{aligned}$$

and from (3.24)

$$\begin{aligned} \frac{{\mathcal {G}}(Z_2)}{\Lambda D_0(Z_2)}=\frac{2k\left( |\lambda _+|+O(a)\right) +O(1)}{|\lambda _+|}>k \end{aligned}$$

for \(0<a<a^*\) small enough and \(k\ge k_1\) large enough. Inserting this into (3.34) yields (3.29). \(\square \)

3.6 Hardy inequality and compactness

We let \(k\ge k_1\) large enough so that (3.30) holds and extend the measure g by zero for \(Z\ge Z_a\). We let \(\chi \) be a smooth cut off function supported strictly inside the light cone \(|Z|<Z_2\) with

$$\begin{aligned} g\ge \frac{1}{2}\ \ \text{ on }\ \ \mathrm{Supp}\chi . \end{aligned}$$

Let

$$\begin{aligned} {\mathcal {D}}_\Phi =\left\{ \Phi \in {\mathcal {C}}^{\infty }([0,Z_a],{\mathbb {C}})\ \ \text{ with } \text{ spherical } \text{ symmetry }\right\} \end{aligned}$$

be the space of test functions and

$$\begin{aligned} \langle \langle \Phi ,{\tilde{\Phi }}\rangle \rangle= & {} -({\mathcal {L}}_g\Phi _k,{\tilde{\Phi }}_{k})_g+\int \chi \Phi \overline{{\tilde{\Phi }}}gZ^{d-1}dZ \end{aligned}$$
(3.35)

be a Hermitian scalar product, where we recall the notation (1.13). We let \({\mathbb {H}}_{\Phi }\) be the completion of \({\mathcal {D}}_\Phi \) for the norm associated to (3.35). We claim the following compactness subcoercivity estimate:

Lemma 3.9

(Subcoercivity estimate) For \(0<\nu <1\):

$$\begin{aligned} \langle \langle \Phi ,\Phi \rangle \rangle\gtrsim & {} \int \frac{|\Phi _k|^2}{Z_a-Z}gZ^{d-1}dZ\nonumber \\&+\sum _{m=0}^{2k} \int |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ. \end{aligned}$$
(3.36)

Furthermore, there exists \(c>0\) and a sequence \(\mu _n>0\) with \(\mathop {\mathrm{lim}}_{n\rightarrow +\infty }\mu _n=+\infty \) and \(\Pi _n\in {\mathbb {H}}_{\Phi }\), \(c_n>0\) such that \(\forall n\ge 0\), \(\forall \Phi \in {\mathbb {H}}_{\Phi }\),

$$\begin{aligned} \langle \langle \Phi ,\Phi \rangle \rangle\ge & {} c\int \frac{|\Phi _k|^2}{Z_a-Z}gZ^{d-1}dZ\nonumber \\&+\mu _n \sum _{m=0}^{2k} \int |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ\nonumber \\&- c_n\sum _{i=1}^n(\Phi ,\Pi _i)_g^2. \end{aligned}$$
(3.37)

Proof

This is a classical Hardy and Sobolev based argument with a loss \(\nu \). We provide a proof for the reader’s convenience.

Step 1 Interior estimate. Let \(Z_0<Z_a\) which will be chosen close enough to \(Z_a\) in step 2. Then, we have

$$\begin{aligned}&\int _0^{Z_0} \frac{|\Phi _k|^2}{Z_a-Z}gZ^{d-1}dZ+\sum _{m=0}^{2k} \int _0^{Z_0} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ\\&\quad \le C_{Z_0}\Vert \Phi \Vert _{H^{2k}(0, Z_0)}^2\\&\quad \le C_{Z_0}\left[ \int _0^{Z_0} |\partial _Z\Phi _k|^2Z^{d-1}dZ+ \int _0^{Z_0} \chi |\Phi (Z)|^2Z^{d-1}dZ\right] . \end{aligned}$$

Since \(-Z^2D_a\) and g are smooth and satisfy \(-Z^2D_a>0\) and \(g>0\) on \([0,Z_0]\), we infer

$$\begin{aligned} \langle \langle \Phi ,\Phi \rangle \rangle\ge & {} c_{Z_0}\left[ \int _0^{Z_0} \frac{|\Phi _k|^2}{Z_a-Z}gZ^{d-1}dZ\right. \\&\left. +\sum _{m=0}^{2k} \int _0^{Z_0} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ\right] \end{aligned}$$

for some \(c_{Z_0}>0\). Thus, to prove (3.36), it remains to consider the region \((Z_0,Z_a)\). This will be done in steps 2 and 3.

Step 2 Hardy inequality with loss. Let \(0<\nu < 1\), we claim the lossy Hardy bound for all \(\Phi \in {\mathcal {D}}_\Phi \):

$$\begin{aligned} \sum _{m=0}^{2k} \int _{Z_0}^{Z_a} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ\le c_{\nu } \langle \langle \Phi ,\Phi \rangle \rangle . \end{aligned}$$
(3.38)

Indeed, let \(Z_0=Z_a-\delta \) with \(\delta >0\) small enough, we estimate by Taylor expansion for \(Z_0\le Z< Z_a\) for \(0\le m\le 2k\):

$$\begin{aligned}&|\partial _Z^{m}\Phi (Z)|^2\le C\left[ \sum _{j=m}^{2k}\left| \partial _Z^{j}\Phi \left( Z_0\right) \right| ^2+\left( \int _{Z_0}^Z\left| \partial _Z^{2k+1}\Phi (\tau )\right| d\tau \right) ^2\right] \end{aligned}$$

From Sobolev,

$$\begin{aligned} \sum _{j=m}^{2k}\left| \partial _Z^{j}\Phi \left( Z_0\right) \right| ^2\le C_{Z_0}\Vert \Phi \Vert _{H^{2k+1}(0, Z_0)}^2\le C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle \end{aligned}$$

and hence

$$\begin{aligned} |\partial _Z^{m}\Phi (Z)|^2\le & {} C_{Z_0} \langle \langle \Phi ,\Phi \rangle \rangle +C\left( \int _{Z_0}^Z\left| \partial _Z\Delta ^k\Phi (\tau )\right| d\tau \right) ^2\\&+C\left( \int \sum _{j=1}^{2k}|\partial _Z^j \Phi |d\tau \right) ^2\\\le & {} C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle +C\left( \int _{Z_0}^Z|\partial _Z\Phi _k|^2(Z_a-Z)^{1-\nu }Z^{d-1}dZ\right) \\&\times \left( \int _{Z_0}^Z\frac{d\tau }{(Z_a-\tau )^{1-\nu }}\right) \\&+C\sum _{j=1}^{2k}\left( \int _{Z_0}^{Z}\frac{|\partial _Z^j\Phi |^2}{(Z_a-Z)^{1-\nu }}dZ\right) \left( \int _{Z_0}^Z(Z_a-\tau )^{1-\nu }d\tau \right) \\\le & {} C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle +C\delta ^\nu \int _{Z_0}^Z|\partial _Z\Phi _k|^2(Z_a-\tau )^{1-\nu }d\tau \\&+C\delta \sum _{j=m}^{2k}\left( \int _{Z_0}^Z\frac{|\partial _Z^j\Phi |^2}{(Z_a-\tau )^{1-\nu }}d\tau \right) , \end{aligned}$$

where we used the fact that \(0<\nu <1\) and \(Z_a-Z_0=\delta \). Using again \(0<\nu <1\) and \(Z_a-Z_0=\delta \), as well as Fubini and the fact that \(\partial _Zg<0\) on \((Z_0, Z_a)\) for \(Z_0\) close enough to \(Z_a\) so that g is decreasing on \((Z_0, Z_a)\), we infer

$$\begin{aligned}&\sum _{m=0}^{2k} \int _{Z_0}^{Z_a} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ\\&\quad \le C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle +C\delta ^\nu \int _{Z_0}^{Z_a}\frac{g}{(Z_a-Z)^{1-\nu }}\\&\qquad \times \left( \int _{Z_0}^Z|\partial _Z\Phi _k|^2(Z_a-\tau )^{1-\nu }d\tau \right) Z^{d-1}dZ\\&\qquad +C\delta \sum _{j=0}^{2k}\int _{Z_0}^{Z_a}\frac{g}{(Z_a-Z)^{1-\nu }}\left( \int _{Z_0}^Z\frac{|\partial _Z^j\Phi |^2}{(Z_a-\tau )^{1-\nu }}d\tau \right) Z^{d-1}dZ\\&\quad \le C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle +C\delta ^\nu \int _{Z_0}^{Z_a}|\partial _Z\Phi _k|^2g(\tau )(Z_a-\tau )^{1-\nu }\\&\qquad \times \left( \int _\tau ^{Z_a}\frac{dZ}{(Z_a-Z)^{1-\nu }}\right) \tau ^{d-1}d\tau \\&\qquad +C\delta \sum _{j=0}^{2k}\int _{Z_0}^{Z_a}\frac{|\partial _Z^j\Phi |^2g(\tau )}{(Z_a-\tau )^{1-\nu }}\left( \int _\tau ^{Z_a}\frac{dZ}{(Z_a-Z)^{1-\nu }}\right) \tau ^{d-1}d\tau \\&\quad \le C_{Z_0}\langle \langle \Phi ,\Phi \rangle \rangle +C\delta ^\nu \int _{Z_0}^{Z_a}|\partial _Z\Phi _k|^2g(\tau )(Z_a-\tau )\tau ^{d-1}d\tau \\&\qquad +C\delta \sum _{j=0}^{2k}\int _{Z_0}^{Z_a}\frac{|\partial _Z^j\Phi |^2g(\tau )}{(Z_a-\tau )^{1-\nu }}\tau ^{d-1}d\tau . \end{aligned}$$

Letting \(\delta =\delta (\nu )\) small enough and estimating from (3.26)

$$\begin{aligned} -(D_a)(Z)\ge c(Z_a-Z) \end{aligned}$$
(3.39)

yields (3.38).

Step 3 Sharp Hardy. We now claim the sharp Hardy inequality for \(f\in {\mathcal {D}}_\Phi \):

$$\begin{aligned} \langle \langle \Phi ,\Phi \rangle \rangle \gtrsim \int _{Z_0}^{Z_a} \frac{|\Phi _k|^2}{(Z_a-Z)}gZ^{d-1}dZ. \end{aligned}$$
(3.40)

Indeed, recall (3.29), (3.30) near \(Z_a\):

$$\begin{aligned} g(Z)=c(Z_a-Z)^{c_g}\left[ 1+O(Z-Z_a)\right] , \end{aligned}$$

then integrating by parts:

$$\begin{aligned}&\int _{Z_0}^{Z_a} \frac{|\Phi _k|^2}{(Z_a-Z)}gZ^{d-1}dZ\lesssim \int _{Z_0}^{Z_a} |\Phi _k|^2(Z_a-Z)^{c_g-1}dZ\\&\quad =-\frac{1}{c_g}[|\Phi _k|^2(Z_a-Z)^{c_g}]_{Z_0}^{Z_a}+\frac{1}{c_g}\int _{Z_0}^{Z_a}2\Phi _k\partial _Z\Phi _k(Z_a-Z)^{c_g}dZ\\&\quad \lesssim |\Phi _k|^2(Z_0)+\left( \int |\Phi _k|^2(Z_a-Z)^{c_g-1}Z^{d-1}dZ\right) ^{\frac{1}{2}}\\&\qquad \times \left( \int |\partial _Z\Phi _k|^2(Z_a-Z)^{c_g+1}Z^{d-1}dZ\right) ^{\frac{1}{2}}\\&\quad \lesssim \langle \langle \Phi ,\Phi \rangle \rangle +\left( \int _{Z_0}^{Z_a} \frac{|\Phi _k|^2}{(Z_a-Z)}gZ^{d-1}dZ\right) ^{\frac{1}{2}}\\&\qquad \times \left( \int |\partial _Z\Phi _k|^2g(-D_a)Z^{d-1}dZ\right) ^{\frac{1}{2}}, \end{aligned}$$

where we used (3.39). The bound (3.40) now follows using Hölder. Together with steps 1 and 2, this concludes the proof of (3.36).

Step 4 Compactness. We now turn to the proof of (3.37) which follows from a standard compactness argument. Let us consider \(T\in L^2_g\). Then from (3.36), the antilinear form

$$\begin{aligned} h \mapsto ( T,h)_g \end{aligned}$$

is continuous on \({\mathbb {H}}_\Phi \), and hence by Riesz Theorem, there exists a unique \(L(T)\in {\mathbb {H}}_\Phi \) such that

$$\begin{aligned} \forall h\in {\mathbb {H}}_\Phi , \ \ \langle \langle L(T), h\rangle \rangle =(T,h)_g, \end{aligned}$$
(3.41)

and the linear map L is bounded from \(L^2_g\) to \({\mathbb {H}}_\Phi \). For any \(0<\delta <Z_a\), we have in view of (3.36)

$$\begin{aligned} \Vert h\Vert _{L^2_g}\le & {} \delta ^{\frac{1-\nu }{2}}\left\| \frac{h}{(Z_a-Z)^{\frac{1-\nu }{2}}}\right\| _{L^2_g}+\Vert h\Vert _{L^2_g(Z\le Z_a-\delta )}\\\lesssim & {} \delta ^{\frac{1-\nu }{2}}\left\| h\right\| _{{\mathbb {H}}_\Phi }+\Vert h\Vert _{L^2_g(Z\le Z_a-\delta )}. \end{aligned}$$

Relying on the smallness of \(\delta ^{\frac{1-\nu }{2}}\) for the first term, and Rellich Theorem for the second one, we easily infer that

$$\begin{aligned} {\mathbb {H}}_\Phi \text { embeds compactly in }L^2_g. \end{aligned}$$
(3.42)

Since L is bounded from \(L^2_g\) to \({\mathbb {H}}_\Phi \), we infer that the map

$$\begin{aligned} L: L^2_g\mapsto L^2_g \end{aligned}$$

is compact. Moreover, if \(\Phi _1=L(T_1),\) \(\Phi _2=L(T_2)\):

$$\begin{aligned} (L(T_1),T_2)_g=(\Phi _1, T_2)_g=\overline{(T_2,\Phi _1)_g}=\overline{\langle \langle LT_2,\Phi _1\rangle \rangle }=\langle \langle \Phi _1,\Phi _2\rangle \rangle \end{aligned}$$

and hence interchanging the roles of \(T_1,T_2\):

$$\begin{aligned} (T_1,L(T_2))_g=\overline{(L(T_2),T_1)_g}=\overline{\langle \langle \Phi _2,\Phi _1\rangle \rangle }=\langle \langle \Phi _1,\Phi _2\rangle \rangle =(L(T_1),T_2)_g \end{aligned}$$

and L is selfadjoint on \(L^2_g\). Since \(L>0\) from (3.41), we conclude that L is a diagonalizable with a non increasing sequences of eigenvalues \(\lambda _n>0\), \(\mathop {\mathrm{lim}}_{n\rightarrow +_\infty }\lambda _n=0\), and let \((\Pi _{n,i})_{1\le i\le I(n)}\) be an \(L^2_g\) orthonormal basis for the eigenvalue \(\lambda _n\). The eigenvalue equation implies \(\Pi _{n,i}\in {\mathbb {H}}_\Phi \).

Let then

$$\begin{aligned} {\mathcal {A}}_n= & {} \left\{ \Phi \in {\mathbb {H}}_\Phi , \ \ \int |\Phi |^2gZ^{d-1}dZ=1, (\Phi ,\Pi _{j,i})_{g}=0, \ \ 1\le i\le I(j), \right. \\&\left. 1\le j\le n\right\} \end{aligned}$$

and the minimization problem

$$\begin{aligned} I_n=\inf _{\Phi \in {\mathcal {A}}_n} \langle \langle \Phi ,\Phi \rangle \rangle , \end{aligned}$$

then the infimum is attained in view of (3.42) at \(\Phi \in {\mathcal {A}}_n\) and, by a standard Lagrange multiplier argument:

$$\begin{aligned} \forall h\in {\mathbb {H}}_\Phi , \ \ \langle \langle \Phi ,h\rangle \rangle =\sum _{j=1}^n\sum _{i=1}^{I(j)}\alpha _{i,j}(\Pi _{j,i},h)_g+\alpha (\Phi , h)_g. \end{aligned}$$

Letting \(h=\Pi _{i,j}\) implies \(\alpha _{i,j}=0\) and hence from (3.41):

$$\begin{aligned} L(\Phi )=\frac{1}{\alpha } \Phi , \end{aligned}$$

which together with our orthogonality conditions implies

$$\begin{aligned} \frac{1}{\alpha }\le \lambda _{n+1} \end{aligned}$$

and hence

$$\begin{aligned} I_n=\langle \langle \Phi ,\Phi \rangle \rangle =\alpha \langle \langle L(\Phi ),\Phi \rangle \rangle =\alpha (\Phi ,\Phi )_g=\alpha \ge \frac{1}{\lambda _{n+1}}. \end{aligned}$$
(3.43)

Also, for \(Z_0=Z_a-\delta \) with \(\delta >0\) small enough, we estimate from (3.38)

$$\begin{aligned} \sum _{m=0}^{2k} \int _{Z_0}^{Z_a} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ \le c_\nu \delta ^{\frac{\nu }{2}}\langle \langle \Phi ,\Phi \rangle \rangle . \end{aligned}$$

On the other hand, from Rellich and an elementary compactness argument, for all \(Z_a>0\), \(\delta >0\), \(\epsilon >0\), \(k\ge 1\), there exists \(c_{Z_a,\delta ,\epsilon ,k}>0\) such that

$$\begin{aligned} \sum _{m=0}^{2k}\int _{Z\le Z_a-\delta }|\partial _Z^m\Phi |^2Z^{d-1}dZ\le & {} \epsilon \int _{Z\le Z_a-\delta }|\partial _Z\Delta ^k\Phi |^2Z^{d-1}dZ\\&+c_{Z_a,\delta ,\epsilon ,k} \int _{Z\le Z_a-\delta }|\Phi |^2Z^{d-1}dZ. \end{aligned}$$

Summing the two inequalities yields for all \(\delta >0\) small and \(\epsilon \) smaller still:

$$\begin{aligned}&\sum _{m=0}^{2k} \int _0^{Z_a} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ \\&\quad \le c_{\nu }\delta ^{\frac{\nu }{2}}\langle \langle \Phi ,\Phi \rangle \rangle +{{\tilde{c}}}_{Z_a,\delta ,k} \int _0^{Z_a}|\Phi |^2gZ^{d-1}dZ. \end{aligned}$$

Together with (3.43), this implies for any \(\Phi \) satisfying the orthogonality conditions \((\Phi ,\Pi _{j,i})_{g}=0, \ \ 1\le i\le I(j), \ \ 1\le j\le n\), and for any \(\delta >0\)

$$\begin{aligned} \sum _{m=0}^{2k} \int _0^{Z_a} |\partial _Z^{m}\Phi (Z)|^2\frac{g}{(Z_a-Z)^{1-\nu }}Z^{d-1}dZ \le \Big (c_\nu \delta ^{\frac{\nu }{2}}+{{\tilde{c}}}_{\delta ,Z_a,k}\lambda _{n+1}\Big )\langle \langle \Phi , \Phi \rangle \rangle , \end{aligned}$$

which yields (3.37). \(\square \)

3.7 Accretivity

We now turn to the proof of the accretivity of the operator \({\mathcal {M}}\).

Hilbert space. Recall (3.35). We define the space of test functions

$$\begin{aligned} {\mathcal {D}}_0={\mathcal {D}}_\Phi \times {{\mathcal {D}}_\Phi }, \end{aligned}$$

and let \({{\mathbb {H}}}_{2k}\) be the completion of \({\mathcal {D}}_0\) for the scalar product:

$$\begin{aligned} \langle X,{\tilde{X}}\rangle = \langle \langle \Phi , {{\tilde{\Phi }}}\rangle \rangle +(T_k, {\tilde{T}}_k)_g+\int \chi T\overline{{\tilde{T}}}Z^{d-1}dZ, \end{aligned}$$
(3.44)

which is a coercive Hermitian form from (3.36).

Unbounded operator. Following (3.15), we define the operator

$$\begin{aligned} {\mathcal {M}}=\left( \begin{array}{cc} -aH_2\Lambda &{}\quad 1\\ (p-1)Q\Delta -(1-a)^2H_2^2\Lambda ^2+\tilde{A_2}\Lambda +A_3&{}\quad -(2-a)H_2\Lambda +A_2\end{array}\right) \end{aligned}$$

with domain

$$\begin{aligned} D({\mathcal {M}})=\{X\in {\mathbb {H}}_{2k}, \ \ {\mathcal {M}} X\in {\mathbb {H}}_{2k}\} \end{aligned}$$
(3.45)

equipped with the domain norm. We then pick suitable directions \((X_i)_{1\le i\le N}\in {\mathbb {H}}_{2k}\) and consider the finite rank projection operator

$$\begin{aligned} {\mathcal {A}}=\sum _{i=1}^N\langle \cdot ,X_i\rangle X_i. \end{aligned}$$

The aim of this section is to prove the following accretivity property:

Proposition 3.10

(Maximal accretivity/dissipativity) Let

$$\begin{aligned} \mu ,\quad r>0. \end{aligned}$$

There exist \(k^*\gg 1\) and \(0<c^*,a^*\ll 1\) such that for all \(k\ge k^*\), \(\forall 0<a<a^*\) small enough, there exist \(N=N(k,a)\) directions \((X_i)_{1\le i\le N}\in {\mathbb {H}}_{2k}\) such that the modified unbounded operator

$$\begin{aligned} \tilde{{\mathcal {M}}}={\mathcal {M}} - {\mathcal {A}} \end{aligned}$$

is dissipative:Footnote 7

$$\begin{aligned} \forall X\in {\mathcal {D}}({\mathcal {M}}), \ \ \mathfrak {R}\langle -\tilde{{\mathcal {M}}} X,X\rangle \ge c^*ak \langle X,X\rangle \end{aligned}$$
(3.46)

and maximal:

$$\begin{aligned} \forall R>0, \ \ \forall F\in {\mathbb {H}}_{2k}, \ \ \exists X\in {\mathcal {D}}({\mathcal {M}})\ \ \text{ such } \text{ that } \ \ (-\tilde{{\mathcal {M}}}+R)X=F. \end{aligned}$$
(3.47)

Remark 3.11

We recall that maximal dissipative operators are closed.

Proof of Proposition 3.10

given \(R>R^*(k)\) large enough, we define the space of test functions

$$\begin{aligned} {\mathcal {D}}_{R}:= & {} \Big \{X=(\Phi , T), \ \ X\in C^{\frac{\sqrt{R}}{2}}([0,Z_a])\times C^{\frac{\sqrt{R}}{2}}([0,Z_a])\Big \}\nonumber \\&\cap \Big \{X \ \ / \ \ (-{\mathcal {M}}+R)X\in C^\infty ([0,Z_a])\times C^\infty ([0,Z_a])\Big \}. \end{aligned}$$
(3.48)

In steps 1–3 below, we prove (3.46) for \(X\in {\mathcal {D}}_R\) so that all integrations by parts in steps 1–3 are justified, and all boundary terms at \(Z=Z_a\) vanish due to the vanishing of g at \(Z=Z_a\). In steps 4 and 5, for any smooth F on \([0,Z_a]\), we show existence and uniqueness of a solution \(X\in {{\mathbb {H}}}_{2k}\) to \((-{\mathcal {M}}+R)X=F\) for \(R>R^*(k)\) large enough. In step 6, we prove that \({\mathcal {D}}_R\) is dense in \(D({\mathcal {M}})\). In step 7, we conclude the proof of (3.46) and (3.47).

Step 1 Main integration by parts. Let \(X\in {\mathcal {D}}_R\) for \(R>R^*(k)\) large enough. We aim at proving (3.46) and split the computation in two:

$$\begin{aligned} \left| \begin{array}{lll} \displaystyle \langle X,{\tilde{X}}\rangle _1=-({\mathcal {L}}_g\Phi _k,{{\tilde{\Phi }}_{k}})_g+(T_k,{\tilde{T}}_k)_g,\\ \displaystyle \langle X,{\tilde{X}}\rangle _3=\int \chi \Phi \overline{ {\tilde{\Phi }}} +\int \chi T\overline{{\tilde{T}}}. \end{array}\right. \end{aligned}$$

In step 1, we consider the principal part. We compute from (3.27):

$$\begin{aligned}&-\mathfrak {R}\langle {\mathcal {M}}X,X\rangle _1=\mathfrak {R}( {\mathcal {L}}_g\Delta ^k({\mathcal {M}}X)_\Phi ,\Phi _k)_g-\mathfrak {R}(\Delta ^k({\mathcal {M}}X)_T,T_k)_g\\&\quad = -\mathfrak {R}\left\{ \int \nabla \left[ -aH_2\Lambda \Phi _k-2ak(H_2+\Lambda H_2)\Phi _k+ T_k+(\widetilde{{\mathcal {M}}_k}X)_\Phi \right] \right. \\&\qquad \left. \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}\mu ^2gdZ \right\} \\&\qquad - \mathfrak {R}\left\{ \int \left[ {\mathcal {L}}_g \Phi _k-(2-a)H_2\Lambda T_k-2k(2-a)(H_2+\Lambda H_2) T_k \right. \right. \\&\qquad \left. \left. +A_2T_k +(\widetilde{{\mathcal {M}}_k}X)_T\right] \overline{T_k}g{Z^{d-1}}dZ\right\} \\&\quad = -\mathfrak {R}\left\{ \int \nabla \left[ -aH_2\Lambda \Phi _k-2ak(H_2+\Lambda H_2)\Phi _k\right. \right. \\&\qquad \left. \left. +(\widetilde{{\mathcal {M}}_k}X)_\Phi \right] \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}g\mu ^2dZ \right\} \\&\qquad - \mathfrak {R}\left\{ \int \left[ -(2-a)H_2\Lambda T_k-2k(2-a)(H_2+\Lambda H_2) T_k \right. \right. \\&\qquad \left. \left. +A_2T_k +(\widetilde{{\mathcal {M}}_k}X)_T\right] \overline{T_k}g{Z^{d-1}}dZ\right\} . \end{aligned}$$

\(T_k\) terms. We use

$$\begin{aligned} \mathfrak {R}\left( \int fh \overline{\Lambda h}\right) =-\frac{1}{2}\int |h|^2 f\left( d+\frac{\Lambda f}{f}\right) \end{aligned}$$

to compute

$$\begin{aligned}&-\mathfrak {R}\left\{ \int \left[ -(2-a)H_2\Lambda T_k\right] \overline{T_k}g{Z^{d-1}}dZ\right\} \\&\quad =-\frac{2-a}{2}\int |T_k|^2gH_2\left( d+\frac{\Lambda g}{g}+\frac{\Lambda H_2}{H_2}\right) \end{aligned}$$

and hence

$$\begin{aligned}&- \mathfrak {R}\left\{ \int \left[ -(2-a)H_2\Lambda T_k-2k(2-a)(H_2+\Lambda H_2) T_k +A_2T_k\right] \overline{T_k}g{Z^{d-1}}dZ\right\} \\&\quad = (2-a)\int A_5 H_2|T_k|^2g{Z^{d-1}}dZ \end{aligned}$$

with

$$\begin{aligned} A_5:= -\frac{1}{2}\left[ d+\frac{\Lambda g}{g}+\frac{\Lambda H_2}{H_2}\right] +2k\left( 1+\frac{\Lambda H_2}{H_2}\right) -\frac{A_2}{(2-a)H_2}. \end{aligned}$$
(3.49)

\(\Phi _k\) terms. We first compute:

$$\begin{aligned}&-\mathfrak {R}\left\{ \int \nabla \left[ -2ak(H_2+\Lambda H_2)\Phi _k\right] \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ \right\} \\&\quad = 2ak\int (H_2+\Lambda H_2)|\nabla \Phi _k|^2Z^2(-D_a)Z^{d-1}gdZ\\&\qquad + 2ak\mathfrak {R}\left\{ \int \Phi _k\nabla (H_2+\Lambda H_2)\cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ \right\} \\&\quad = 2ak\int (H_2+\Lambda H_2)|\nabla \Phi _k|^2Z^2(-D_a)Z^{d-1}gdZ\\&\qquad - ak\int |\Phi _k|^2\nabla \cdot \left( Z^2(-D_a)\nabla (H_2+\Lambda H_2) g\right) Z^{d-1}dZ. \end{aligned}$$

For the second term:

$$\begin{aligned}&-\mathfrak {R}\left\{ \int \nabla \left[ -aH_2\Lambda \Phi _k\right] \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ\right\} \\&\quad = -a\mathfrak {R}\left\{ \int \partial _Z(H_2\Lambda \Phi _k)H_2\overline{\Lambda \Phi _k} \frac{D_aZ^{d}}{H_2} g dZ\right\} \\&\quad = \frac{a}{2}\int |H_2\Lambda \Phi _k|^2\frac{D_aZ^{d}g}{H_2}\left( \frac{\partial _ZD_a}{D_a}+\frac{d}{Z}-\frac{\partial _ZH_2}{H_2}+\frac{\partial _Z g}{g}\right) dZ\\&\quad = -\frac{a}{2}\int |\partial _Z\Phi _k|^2H_2\left( \frac{\Lambda D_a}{D_a}+d-\frac{\Lambda H_2}{H_2}+\frac{\Lambda g}{g}\right) (-D_a) gZ^2Z^{d-1}dZ. \end{aligned}$$

We have therefore obtained the formula:

$$\begin{aligned} -\mathfrak {R}\langle {\mathcal {M}}X,X\rangle _1= & {} (2-a)\int A_5H_2|T_k|^2 g+\mu ^2a \nonumber \\&\times \int |\nabla \Phi _k|^2A_6Z^2(-D_a)Z^{d-1}gdZ \nonumber \\&- \mu ^2ak\int |\Phi _k|^2\nabla \cdot \left( Z^2(-D_a)\nabla (H_2+\Lambda H_2) g \right) Z^{d-1}dZ \nonumber \\&- \mu ^2\mathfrak {R}\left\{ \int \nabla (\widetilde{\mathcal {M}_k}X)_\Phi \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ\right\} \nonumber \\&- \mathfrak {R}\left\{ \int (\widetilde{\mathcal {M}_k}X)_T\overline{T_k}g{Z^{d-1}}dZ\right\} \end{aligned}$$
(3.50)

where we have defined

$$\begin{aligned} A_6 = 2k(H_2+\Lambda H_2)-\frac{H_2}{2}\left( \frac{\Lambda D_a}{D_a}+d-\frac{\Lambda H_2}{H_2}+\frac{\Lambda g}{g}\right) . \end{aligned}$$

We now claim the following lower bounds on \(A_5,A_6\): there exist universal constants \(k^*\gg 1\), \(0<c^*,a^*\ll 1\) such that for all \(k\ge k^*\) and \(0<a<a^*\),

$$\begin{aligned} \forall 0\le Z\le Z_1, \ \ \left| \begin{array}{l} A_5 \ge \frac{c^*k}{Z_a-Z},\\ A_6 \ge \frac{c^*k}{Z_a-Z}. \end{array}\right. \end{aligned}$$
(3.51)

Proof of (3.51). Recall (3.32), (3.33):

$$\begin{aligned} -\frac{\Lambda g}{g}= & {} \frac{1}{(-D_a)}\Big \{-(d-1)\sigma ^2+(1-w)^2-(d+1)D_a-\Lambda D_a \\&+4k\left[ (1-a)^2(1-w) -\sigma F-(1-a)^2(1-w)(w+\Lambda w)\right] -\frac{{\tilde{A}}_2}{\mu ^2}\Big \}\\= & {} \frac{4k}{(-D_a)}\left[ (1-w)-\sigma F-(1-w)(w+\Lambda w)+O\left( a+\frac{1}{k}\right) \right] \end{aligned}$$

and hence from (3.49):

$$\begin{aligned} A_5= & {} -\frac{1}{2}\left[ d+\frac{\Lambda g}{g}+\frac{\Lambda H_2}{H_2}\right] +2k\left( 1+\frac{\Lambda H_2}{H_2}\right) -\frac{A_2}{(2-a)H_2}\\&\quad = \frac{2k}{(-D_a)}\left[ (1-w)-\sigma F-(1-w)(w+\Lambda w)+O\left( a+\frac{1}{k}\right) \right] \\&\qquad +2k\left( 1-\frac{\Lambda w}{1-w}+O\left( \frac{1}{k}\right) \right) \\&\quad = \frac{2k}{(-D_a)}\left[ (1-w)-\sigma F-(1-w)(w+\Lambda w)\right. \\&\qquad \left. +(-D_a)\left( 1-\frac{\Lambda w}{1-w}\right) +O\left( a+\frac{1}{k}\right) \right] \\&\quad = \frac{2k}{(-D_a)}\left[ (1-w)-\sigma F-(1-w)(w+\Lambda w)\right. \\&\qquad \left. +(-\Delta )\left( 1-\frac{\Lambda w}{1-w}\right) +O\left( a+\frac{1}{k}\right) \right] . \end{aligned}$$

We now compute for \(Z\le Z_2\)

$$\begin{aligned}&(1-w)-\sigma F-(1-w)(w+\Lambda w)+(-\Delta )\left( 1-\frac{\Lambda w}{1-w}\right) \\&\quad = (1-w)(1-w-\Lambda w)-\sigma F+(\sigma ^2-(1-w)^2)\frac{1-w-\Lambda w}{1-w}\\&\quad = \frac{\sigma ^2(1-w-\Lambda w)}{1-w}-\sigma F= \frac{\sigma ^2}{1-w}\left[ 1-w-\Lambda w-\frac{1-w}{\sigma }F\right] \\&\quad \ge c\sigma ^2 \end{aligned}$$

from the fundamental coercivity bound (2.21), and hence for \(Z\le Z_a\) and \(a<a^*\) small enough:

$$\begin{aligned} A_5\ge \frac{kc\sigma ^2}{(-D_a)}\ge \frac{kc^*}{Z_a-Z} \end{aligned}$$

for some \(c^*\) independent of ka. Similarly:

$$\begin{aligned} A_6= & {} \frac{2kH_2}{-D_a}\left\{ \left[ 1+\frac{\Lambda H_2}{H_2}+O\left( \frac{1}{k}\right) \right] \right. \\&\quad \left. (-D_a)+(1-w)-\sigma F-(1-w)(w+\Lambda w)+O\left( a+\frac{1}{k}\right) \right\} \\= & {} \frac{2k\mu (1-w)}{(-D_a)}\left[ (1-w)-\sigma F-(1-w)(w+\Lambda w)\right. \\&\quad \left. +(-\Delta )\left( 1-\frac{\Lambda w}{1-w}\right) +O\left( a+\frac{1}{k}\right) \right] \\\ge & {} \frac{kc^*}{Z_a-Z} \end{aligned}$$

arguing as above. This concludes the proof of (3.51).

Step 2 No derivatives term. We compute

$$\begin{aligned} -\mathfrak {R}\langle {\mathcal {M}}X,X\rangle _3= & {} -\mathfrak {R}\left\{ \int \chi ({\mathcal {M}}X)_\Phi {\overline{\Phi }}Z^{d-1}dZ\right. \nonumber \\&\left. +\int \chi ({\mathcal {M}}X)_T{\overline{T}}Z^{d-1}dZ\right\} \nonumber \\= & {} -\mathfrak {R}\left\{ \int \chi \left[ -aH_2\Lambda \Phi +T\right] {\overline{\Phi }}Z^{d-1}dZ\right\} \nonumber \\&-\mathfrak {R}\left\{ \int \chi \left[ (p-1)Q\Delta \Phi -(1-a)^2H_2^2\Lambda ^2\Phi +\tilde{A_2}\Lambda \Phi \right. \right. \nonumber \\&\left. \left. +A_3\Phi -(2-a)H_2\Lambda T +A_2T\right] {\overline{T}}\right\} \nonumber \\= & {} O\left( \int (\chi +|\Lambda \chi |)\left( |\Phi |^2+|\partial _Z\Phi |^2\right. \right. \nonumber \\&\left. \left. +|\partial _Z^2\Phi |^2+|T|^2\right) \right) .\nonumber \\ \end{aligned}$$
(3.52)

Step 3 Accretivity in \({\mathcal {D}}_0\). We compute from (3.52), (3.50):

$$\begin{aligned}&-\mathfrak {R}\langle {\mathcal {MX}},X\rangle = -\mathfrak {R}\langle {\mathcal {MX}},X\rangle _1 -\mathfrak {R}\langle {\mathcal {MX}},X\rangle _3\\&\quad = (2-a)\int A_5H_2|T_k|^2 g+\mu ^2a\int |\nabla \Phi _k|^2A_6Z^2(-D_a)Z^{d-1}gdZ\\&\qquad - \mu ^2ak\int |\Phi _k|^2\nabla \cdot \left( Z^2(-D_a)\nabla (H_2+\Lambda H_2) g \right) Z^{d-1}dZ\\&\qquad - \mu ^2\mathfrak {R}\left\{ \int \nabla (\widetilde{{\mathcal {M}}_k}X)_\Phi \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ\right\} \\&\qquad - \mathfrak {R}\left\{ \int (\widetilde{{\mathcal {M}}_k}X)_T\overline{T_k}g{Z^{d-1}}dZ\right\} \\&\qquad + O\left( \int (\chi +|\Lambda \chi |)\left( |\Phi |^2+|\partial _Z\Phi |^2+|\partial _Z^2\Phi |^2+|T|^2\right) \right) . \end{aligned}$$

We lower bound from (3.51) and the fact that \(H_2\gtrsim 1\):

$$\begin{aligned}&(2-a)\int A_5H_2|T_k|^2 g+\mu ^2a\int |\nabla \Phi _k|^2A_6Z^2(-D_a)Z^{d-1}g\,dZ\\&\quad \ge c^*ak\left[ \int \left( \frac{|T_k|^2}{Z_a-Z}+|\nabla \Phi _k|^2\frac{Z^2(-D_a)}{Z_a-Z}\right) gZ^{d-1}dZ\right] . \end{aligned}$$

The smoothness and boundedness of the profile together with (3.32), (3.33) ensure that

$$\begin{aligned} \left| \nabla \cdot \left[ Z^2(-D_a)\nabla (H_2+\Lambda H_2)g \right] \right| \le C_k\frac{Z^2(-D_a)g}{Z_a-Z}\le C_k g\end{aligned}$$

and in view of (3.28),

$$\begin{aligned}&\Bigg | - \mathfrak {R}\left\{ \int \nabla (\widetilde{{\mathcal {M}}_k}X)_\Phi \cdot \overline{\nabla \Phi _k} Z^2(-D_a)Z^{d-1}gdZ\right\} \\&\quad - \mathfrak {R}\left\{ \int (\widetilde{\mathcal {M}_k}X)_T\overline{T_k}g{Z^{d-1}}dZ\right\} \\&\le C_k\left( \int |\nabla \Phi _k|^2Z^2(-D_a)gZ^{d-1}dZ\right) ^\frac{1}{2}\left( \sum _{j=0}^{2k}\int |\partial _Z^j\Phi |^2gZ^{d-1}dZ\right) ^{\frac{1}{2}}\\&\quad +C_k\left( \int |T_k|^2gZ^{d-1}dZ\right) ^\frac{1}{2}\left[ \left( \sum _{j=0}^{2k-1}\int |\partial ^j_ZT|^2gZ^{d-1}dZ\right) ^{\frac{1}{2}}\right. \\&\quad \left. +\left( \sum _{j=0}^{2k}\int |\partial _Z^j\Phi |^2gZ^{d-1}dZ\right) ^{\frac{1}{2}}\right] \end{aligned}$$

The collection of above bounds yields:

$$\begin{aligned} -\mathfrak {R}\langle {\mathcal {MX}},X\rangle\ge & {} c^*ak\left[ \int \frac{|T_k|^2}{Z_a-Z}gZ^{d-1}dZ\right. \\&\left. +\int |\nabla \Phi _k|^2\frac{Z^2(-D_a)}{Z_a-Z}gZ^{d-1}\,dZ\right] \\&- C_k\left[ \sum _{j=0}^{2k}\int |\partial _Z^j\Phi |^2gZ^{d-1}dZ+\sum _{j=0}^{2k-1}\int |\partial ^j_ZT|^2gZ^{d-1}dZ\right] . \end{aligned}$$

We conclude using (3.37) with \(N=N(a,k)\) large enough and its analogue for T:

$$\begin{aligned} -\mathfrak {R}\langle {\mathcal {M}}X,X\rangle \ge c^*ak\langle X,X\rangle -C_{a,k}\sum _{i=1}^N\Big ((\Phi ,\Pi _i)^2_g+(T,{\mathcal {T}}_i)_g^2\Big ). \end{aligned}$$

Therefore,

$$\begin{aligned} -\mathfrak {R}\langle ({{\mathcal {M}}}-{\mathcal {A}})X,X\rangle\ge & {} c^*ak\langle X,X\rangle + \sum _{i=1}^{N}\left( \langle X,X_{i,1}\rangle ^2 +\langle X,X_{i,2}\rangle ^2\right. \\&\left. -C_{a,k}\left[ (\Phi ,\Pi _i)^2_g+(T,{\mathcal {T}}_i)_g^2\right] \right) . \end{aligned}$$

The linear from

$$\begin{aligned} X=(\Phi ,T)\mapsto \sqrt{C_{a,k}}(\Phi ,\Pi _i)_g \end{aligned}$$

from \(({\mathbb {H}}_{2k},\langle \cdot \rangle )\) into \({\mathbb {C}}\) is continuous from Cauchy–Schwarz and (3.36), and hence by Riesz theorem, there exists \(X_i\in {\mathbb {H}}_{2k}\) such that

$$\begin{aligned} \forall X\in {\mathbb {H}}_{2k}, \quad \langle X,X_{i}\rangle =(\Phi ,\Pi _i)_g, \end{aligned}$$

and similarly for \({\mathcal {T}}_i\), and (3.46) follows for \(X\in {\mathcal {D}}_R\).

Step 4 ODE formulation of maximality. Our goal, in steps 4–6, is to prove that for all \(R>0\) large enough,

$$\begin{aligned} \forall F\in C^{\infty }([0,Z_a]), \quad \exists !\, X\in {{\mathbb {H}}}_{2k}\quad \text{ such } \text{ that } \quad (-{\mathcal {M}}+ R)X=F. \end{aligned}$$
(3.53)

(3.53) corresponds to solving

$$\begin{aligned} \left| \begin{array}{ll} -\left[ -aH_2\Lambda \right] \Phi -T+R\Phi =F_\Phi ,\\ -\left\{ \left[ (p-1)Q\Delta -(1-a)^2H_2^2\Lambda ^2+\tilde{A_2}\Lambda +A_3\right] \Phi \right. \\ \quad \left. -(2-a)H_2\Lambda T +A_2T\right\} +RT=F_T. \end{array}\right. \end{aligned}$$

Solving for T:

$$\begin{aligned} T= & {} (aH_2\Lambda +R)\Phi -F_\Phi , \end{aligned}$$
(3.54)

we look for \(\Phi \)—solution to the second order elliptic equation:

$$\begin{aligned}&\left[ (p-1)Q\Delta -(1-a)^2H_2^2\Lambda ^2+\tilde{A_2}\Lambda +A_3\right] \Phi \\&\qquad \Big [-(2-a)H_2\Lambda +A_2\Big ]\left[ aH_2\Lambda \Phi +R\Phi -F_\Phi \right] \\&\quad = -F_T+R\left( aH_2\Lambda \Phi +R\Phi -F_\Phi \right) \end{aligned}$$

i.e.

$$\begin{aligned}&(p-1)Q\Delta \Phi -H_2^2\Lambda ^2\Phi +\Lambda \Phi \left[ {\tilde{A}}_2 +aH_2A_2-2RH_2\right. \\&\qquad \left. -a(2-a)H_2\Lambda H_2\right] +(A_3 +RA_2 -R^2)\Phi \\&\quad = -F_T-RF_\Phi +\Big [-(2-a)H_2\Lambda +A_2\Big ]F_\Phi . \end{aligned}$$

Now, we have

$$\begin{aligned} (p-1)Q\Delta \Phi -H_2^2\Lambda ^2\Phi= & {} \Big ((p-1)Q -H_2^2Z^2\Big )\partial _Z^2\Phi \\&\quad + \left( \frac{(d-1)(p-1)Q}{Z} -H_2^2Z\right) \partial _Z\Phi \end{aligned}$$

and hence

$$\begin{aligned}&\Big ((p-1)Q -H_2^2Z^2\Big )\partial _Z^2\Phi +\Bigg \{\frac{(d-1)(p-1)Q}{Z} -H_2^2Z\\&\qquad +Z\left[ {\tilde{A}}_2+aH_2A_2- 2RH_2-a(2-a)H_2\Lambda H_2\right] \Bigg \}\partial _Z\Phi \\&\qquad +(A_3 +RA_2 -R^2)\Phi \\&\quad = -F_T-RF_\Phi +\Big [-(2-a)H_2\Lambda +A_2\Big ]F_\Phi . \end{aligned}$$

Since \((p-1)Q=\mu ^2Z^2\sigma ^2\), we have

$$\begin{aligned}&\Big ((p-1)Q -H_2^2Z^2\Big )\partial _Z^2\Phi +\Bigg \{\frac{(d-1)(p-1)Q}{Z} -H_2^2Z\\&\qquad +Z\left[ {\tilde{A}}_2+aH_2A_2- 2RH_2-a(2-a)H_2\Lambda H_2\right] \Bigg \}\partial _Z\Phi \\&\quad = \Big (\mu ^2\sigma ^2 -H_2^2\Big )Z^2\partial _Z^2\Phi +\Bigg \{(d-1)\mu ^2Z\sigma ^2 -H_2^2Z\\&\qquad +Z\left[ {\tilde{A}}_2+aH_2A_2- 2RH_2-a(2-a)H_2\Lambda H_2\right] \Bigg \}\partial _Z\Phi \\&\qquad = \frac{1}{Z^{d-1}\varpi }\partial _Z\left( Z^{d-1}\varpi \Big (\mu ^2\sigma ^2 -H_2^2\Big )Z^2\partial _Z\Phi \right) \end{aligned}$$

with

$$\begin{aligned}&\left( \frac{\partial _Z\varpi }{\varpi }+\frac{d-1}{Z}\right) \Big (\mu ^2\sigma ^2 -H_2^2\Big )Z^2 \\&\qquad +2Z\Big (\mu ^2\sigma ^2 -H_2^2\Big )+\Big (2\mu ^2\sigma \partial _Z\sigma -2H_2\partial _ZH_2\Big )Z^2\\&\quad = (d-1)\mu ^2Z\sigma ^2 -H_2^2Z\\&\qquad +Z\left[ {\tilde{A}}_2+aH_2A_2-2RH_2-a(2-a)H_2\Lambda H_2\right] , \end{aligned}$$

i.e.

$$\begin{aligned} \frac{\partial _Z\varpi }{\varpi }= & {} -\frac{2}{Z}-\frac{2\mu ^2\sigma \partial _Z\sigma -2H_2\partial _ZH_2}{\mu ^2\sigma ^2 -H_2^2}\\&- \frac{2RH_2-(d-2)H_2^2 -{\tilde{A}}_2-aH_2A_2+a(2-a)H_2\Lambda H_2}{\Big (\mu ^2\sigma ^2 -H_2^2\Big )Z}. \end{aligned}$$

Recalling \(H_2=\mu (1-w)\) yields

$$\begin{aligned} \frac{\partial _Z\varpi }{\varpi }= & {} -\frac{2}{Z}-\frac{\partial _Z[\sigma ^2 -(1-w)^2]}{\sigma ^2 -(1-w)^2}\\&\quad - \frac{\frac{2(1-w)}{\mu }R -(d-2)(1-w)^2 -\frac{{\tilde{A}}_2}{\mu ^2}- a(1-w)\frac{A_2}{\mu } -a(2-a)(1-w)\Lambda w}{Z\Big (\sigma ^2 -(1-w)^2\Big )}. \end{aligned}$$

We therefore define

$$\begin{aligned} \varpi (Z)=\left| \begin{array}{ll}\displaystyle \frac{e^{-F_-(Z)}}{Z^2(\sigma ^2 -(1-w)^2)} \ \ \ \ \text{ for }\ \ 0\le Z\le Z_2,\\ \displaystyle \frac{e^{-F_+(Z)}}{Z^2(\sigma ^2 -(1-w)^2)} \ \ \ \ \text{ for }\ \ Z>Z_2. \end{array}\right. \end{aligned}$$
(3.55)

whereFootnote 8

$$\begin{aligned} F_-(Z)= & {} \int _{\frac{Z_2}{2}}^Z\frac{\frac{2(1-w)}{\mu }R -(d-2)(1-w)^2 -\frac{{\tilde{A}}_2}{\mu ^2} - a(1-w)\frac{A_2}{\mu } -a(2-a)(1-w)\Lambda w}{Z'\Big (\sigma ^2 -(1-w)^2\Big )}dZ' + C_-,\\ F_+(Z)= & {} \int _{2Z_2}^Z\frac{\frac{2(1-w)}{\mu }R -(d-2)(1-w)^2 -\frac{{\tilde{A}}_2}{\mu ^2} - a(1-w)\frac{A_2}{\mu } -a(2-a)(1-w)\Lambda w}{Z'\Big (\sigma ^2 -(1-w)^2\Big )}dZ'+C_+. \end{aligned}$$

In view of the above, we have obtained the elliptic equation:

$$\begin{aligned} \left| \begin{array}{ll} -\frac{1}{Z^{d-1}\varpi }\partial _Z\left( Z^{d-1}\varpi \Big (\sigma ^2 -(1-w)^2\Big )Z^2\partial _Z\Phi \right) \\ \quad +\frac{1}{\mu ^2}(R^2 -A_2R -A_3)\Phi =H,\\ H= \frac{1}{\mu ^2}\left\{ F_T+RF_\Phi +\Big [(2-a)H_2\Lambda -A_2\Big ]F_\Phi \right\} , \end{array}\right. \end{aligned}$$
(3.56)

with T recovered by (3.54). As \(Z\rightarrow Z_2\), we have from (3.24):

$$\begin{aligned} \Delta (Z)=\frac{|\lambda _+|}{Z_2}(Z-Z_2)+O((Z-Z_2)^2) \end{aligned}$$

and hence

$$\begin{aligned} Z(\sigma ^2-(1-w)^2)=|\lambda _+|(Z_2-Z)\left[ 1+O(Z-Z_2)\right] \end{aligned}$$

and hence

$$\begin{aligned}&\frac{\frac{2(1-w)}{\mu }R -(d-2)(1-w)^2 -\frac{{\tilde{A}}_2}{\mu ^2}- a(1-w)\frac{A_2}{\mu } -a(2-a)(1-w)\Lambda w}{Z\Big (\sigma ^2 -(1-w)^2\Big )}\\&\quad =\frac{\frac{2\sigma _2}{{\mu }|\lambda _+|}R\left[ 1+O\left( \frac{1}{R}\right) \right] }{(Z_2-Z)\left[ 1+O(Z-Z_2)\right] }. \end{aligned}$$

Since the profile passes through P2 in a \({\mathcal {C}}^\infty \) way, we obtain the development of the measure at P2: for any \(M\ge 1\),

$$\begin{aligned} \varpi (Z)=|Z_2-Z|^{c_{\varpi }}\left[ 1+\sum _{m=0}^{M}d_{\sigma ,m,R}(Z_2-Z)^{m}+O_M\Big (|Z_2-Z|^{M+1})\right] ,\nonumber \\ \end{aligned}$$
(3.57)

where

$$\begin{aligned} c_{\varpi }=\frac{2\sigma _2}{{\mu }|\lambda _+|}R\left[ 1+O\left( \frac{1}{R}\right) \right] \ge c^*R>0 \end{aligned}$$
(3.58)

for \(R>R^*\) large enough. Note that the above choice of \(C_\pm \) is made to fix the normalization constant in front of \(|Z_2-Z|^{c_{\varpi }}\) to be equal to 1.

Step 5 Solving (3.56). We analyze the singularity of (3.56) at P2 using a change of variables.

\(0\le Z<Z_2\). We let

$$\begin{aligned} \Phi (Z)=\Psi (Y), \ \ Y=h(Z), \ \ h(Z)=\int _{\frac{Z_2}{2}}^Z\frac{dZ'}{{Z'}^{d-1}\varpi {Z'}^2(\sigma ^2-(1-w)^2)}, \end{aligned}$$

which maps (3.56) onto:

$$\begin{aligned} \left| \begin{array}{ll}-\partial _Y^2\Psi +\frac{1}{\mu ^2}(R^2 -A_2R -A_3)Z^{2d}\varpi ^2(\sigma ^2-(1-w)^2)\Psi ={\tilde{H}},\\ {\tilde{H}}=Z^{2d}\varpi ^2(\sigma ^2-(1-w)^2)H. \end{array}\right. \end{aligned}$$
(3.59)

From (3.57),

$$\begin{aligned}&Y=h(Z)=\int _{\frac{Z_2}{2}}^Z\frac{dz}{{z}^{d-1}\varpi {z}^2(\sigma ^2-(1-w)^2)} \nonumber \\&\quad = \int _{\frac{Z_2}{2}}^Z\frac{dz}{{z}^{d-1}{z}|\lambda _+|(Z_2-z)(Z_2-z)^{c_{\varpi }}\left[ 1+\sum _{m=0}^{M}d_{\sigma ,m,R}(Z_2-z)^{m}+O_M\Big (|Z_2-z|^{M+1})\right] } \nonumber \\&\quad =\frac{C}{R\varpi }[1+\Gamma (Z)], \end{aligned}$$
(3.60)

where from (3.58) constant \(C>0\) is independent of R and, choosing \(M=\sqrt{R}\),

$$\begin{aligned} \Gamma (Z)=\sum _{m=1}^{\sqrt{R}}{{\tilde{d}}}_{\sigma ,m,R}(Z_2-Z)^{m}+O\Big ((Z_2-Z)^{\sqrt{R}+1}\Big ) \end{aligned}$$
(3.61)

with similar estimates for derivatives. Hence the potential term in (3.59) can be expanded in Y and estimated as \(Y\rightarrow +\infty \) for R large enough:

$$\begin{aligned}&\frac{1}{\mu ^2}(R^2 -A_2R -A_3)Z^{2d}\varpi ^2(\sigma ^2-(1-w)^2) \nonumber \\&\quad =\frac{C_R}{Y^{2+c_R}}\left[ 1+\sum _{j=1}^{\sqrt{R}}\frac{\tilde{\tilde{d_{j}}}}{Y^{jc_R}}+O\left( \frac{1}{Y^{c_R(\sqrt{R}+1)}}\right) \right] \end{aligned}$$
(3.62)

for some universal constants \(\tilde{\tilde{d_{j}}}\),

$$\begin{aligned} C_R=C+O\left( \frac{1}{R}\right) , \ \ 0<c_R=\frac{1}{c_\varpi }\lesssim \frac{1}{R} \end{aligned}$$

where \(C>0\) is independent of R. Therefore, by an elementary fixed point argument, (3.59) with \({\tilde{H}}=0\) admits a basis of solutions \(\Psi ^-_1\) and \(\Psi ^-_2\) with the following behavior as \(Y\rightarrow +\infty \)

$$\begin{aligned} \left| \begin{array}{l} \Psi ^-_1=1+\sum _{j=1}^{\sqrt{R}}\frac{c_{j,1}}{Y^{jc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \\ \Psi ^-_2=Y\left[ 1+\sum _{j=1}^{\sqrt{R}}\frac{c_{j,2}}{Y^{jc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \right] \end{array}\right. \end{aligned}$$
(3.63)

with similar estimates for derivatives. The sequences \((c_{j,1})_{j=1,2}\) are uniquely determined inductively from (3.59) with \({\tilde{H}}=0\) using the expansion of the potential (3.62).

\(Z_2<Z\le Z_a\). To the right of P2, we let

$$\begin{aligned} \Phi (Z)=\Psi (Y), \ \ Y=h(Z), \ \ h(Z)=\int _{2Z_2}^{Z}\frac{dz}{{z}^{d-1}\varpi {z}^2(\sigma ^2-(1-w)^2)}+{{\tilde{C}}}_+, \end{aligned}$$

which sendsFootnote 9\(Y\rightarrow +\infty \) as \(Z\downarrow Z_2\). We construct a similar basis of homogenous solutions \(\Psi ^+_1\) and \(\Psi ^+_2\) as \(Y\rightarrow +\infty \) with asymptotics given by

$$\begin{aligned} \Psi ^+_1= & {} 1+\sum _{j=1}^{\sqrt{R}}\frac{c_{j,1}}{Y^{jc_R}}+O\left( \frac{1}{Y^{(1+\sqrt{R})c_R}}\right) ,\\ \Psi ^+_2= & {} Y\left[ 1+\sum _{j=1}^{\sqrt{R}}\frac{c_{j,2}}{Y^{jc_R}}+O\left( \frac{1}{Y^{(1+\sqrt{R})c_R}}\right) \right] \end{aligned}$$

with the sequences \(c_{j,1}\), \(c_{j,2}\) the same as in (3.63).

Basis of fundamental solutions. The function \(\Phi _1(Z)=\Psi ^-_1(Y)\) for \(Z<Z_2\) and \(\Phi _1(Z)=\Psi ^+_1(Y)\) for \(Z>Z_2\),obtained by gluing \(\Psi ^\pm _1(Y)\) belongs to \({\mathcal {C}}^{\sqrt{R}}([0,Z_a])\) and is a solution to the homogeneous Eq. (3.59). Let now \(\Phi _{\mathrm{rad}}(Z)\) be the radial solution to the homogeneous problem associated to (3.56) with \(\Phi _{\mathrm{rad}}(0)=1\). Then the wronskian is given by

$$\begin{aligned} W=\partial _Z\Phi _1\Phi _{\mathrm{rad}}-\partial _Z\Phi _{\mathrm{rad}}\Phi _1=\frac{W_0}{Z^{d-1}\varpi Z^2(\sigma ^2-(1-w)^2)}, \end{aligned}$$

where \(W_0\) is a constant. We claim \(W_0\ne 0\). Indeed, otherwise \(\Phi _{\mathrm{rad}}\) is proportionate to \(\Phi _1\) and hence is \(C^{\sqrt{R}}\) on \([0,Z_a]\). In particular, if \(T_{\mathrm{rad}}\) is given by (3.54) with \(F_\Phi =0\), then \(X_{\mathrm{rad}}=(\Phi _{\mathrm{rad}}, T_{\mathrm{rad}})\) satisfies

$$\begin{aligned} (-{\mathcal {M}}+ R)X_{\mathrm{rad}}=0\text { on }(0,Z_a). \end{aligned}$$

Since \(X_{rad}\) is \(C^{\sqrt{R}-1}[[0,Z_2])\), we may apply the analysis in steps 1–4 for \(R>R^*(k)\) large enough and (3.46) holds for \(X_{\mathrm{rad}}\), i.e.

$$\begin{aligned} 0= & {} \mathfrak {R}\langle (-{\mathcal {M}}+ R )X_{rad},X_{rad}\rangle \\= & {} \mathfrak {R}\langle (-{\mathcal {M}} +{\mathcal {A}})X_{rad},X_{rad}\rangle -\mathfrak {R}\langle {\mathcal {A}} X_{rad},X_{rad}\rangle +R\Vert X_{rad}\Vert _{{{\mathbb {H}}}_{2k}}^2\\\ge & {} R\Vert X_{rad}\Vert _{{{\mathbb {H}}}_{2k}}^2 -\langle {\mathcal {A}} X_{rad},X_{rad}\rangle \end{aligned}$$

so that for \(R>R^*(k)\) sufficiently large

$$\begin{aligned} \frac{R}{2}\Vert X_{rad}\Vert _{{{\mathbb {H}}}_{2k}}^2\le 0 \end{aligned}$$

and hence \(X_{rad}=0\) a contradiction. This concludes the proof of \(W_0\ne 0\).

Inner solution of the inhomogeneous problem. \((\Phi _{\mathrm{rad}}, \Phi _1)\) is then a basis for the homogeneous problem corresponding to (3.56). As a consequence, the only solution to (3.56) which is \(o((Z_2-Z)^{-\frac{1}{c_R}})\) at \(Z=Z_2\) is given byFootnote 10

$$\begin{aligned} \Phi (Z)=-\Phi _1(Z)\int _{0}^{Z}\frac{H(\tau )\Phi _{\mathrm{rad}}(\tau )}{W(\tau )}d\tau -\Phi _{\mathrm{rad}}(Z)\int _{Z}^{Z_2}\frac{H(\tau )\Phi _1(\tau )}{W(\tau )}d\tau . \end{aligned}$$

For a smooth H, \(\Phi \) is smooth on \([0,Z_2)\) and we study its regularity at \(Z_2\). In Y variables we obtain for some \(Y_0\) large enough:

$$\begin{aligned} \Psi (Y)= & {} c_{Y_0,H}\Psi ^-_1(Y)-\Psi ^-_1(Y)\int _{Y_0}^{Y}{\tilde{H}}(\tau )\Psi ^-_2(\tau )d\tau \nonumber \\&-\Psi ^-_2(Y)\int _{Y}^{+\infty }{\tilde{H}}(\tau )\Psi ^-_1(\tau )d\tau . \end{aligned}$$
(3.64)

We have from (3.57), (3.60):

$$\begin{aligned} (RY)^{c_R}= \frac{1}{Z_2-Z} \left( \sum _{m=0}^{\sqrt{R}} \beta _m (Z_2-Z)^m + O(|Z_2-Z|^{\sqrt{R}+1})\right) , \end{aligned}$$

and hence

$$\begin{aligned} Z_2-Z = \sum _{m=1}^{\sqrt{R}} \frac{y_m}{Y^{mc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \end{aligned}$$

with similar estimates for derivatives. In particular, a smooth function H(Z) yields expansion for \({{\tilde{H}}}(Z)\):

$$\begin{aligned} {{\tilde{H}}}= & {} (Z_2-Z)^{1+2c^{-1}_R}\left( \sum _{m=0}^{\sqrt{R}} h_m (Z_2-Z)^m+O\Big ((Z_2-Z)^{\sqrt{R}+1}\Big )\right) \\= & {} \sum _{m=1}^{\sqrt{R}} \frac{q_m}{Y^{2+mc_R}}+O\left( \frac{1}{Y^{2+(\sqrt{R}+1)}c_R}\right) . \end{aligned}$$

Conversely, an expansion of the form

$$\begin{aligned} G=\sum _{m=0}^{\sqrt{R}-1} \frac{b_m}{Y^{mc_R}}+O\left( \frac{1}{Y^{\sqrt{R}c_R}}\right) \end{aligned}$$

defines a \(C^{\sqrt{R}}\) function G(Z) at \(Z=Z_2\). Plugging in the asymptotic expansion for \(\Psi ^-_1, \Psi ^-_2\) and \({{\tilde{H}}}\) in (3.64) yields

$$\begin{aligned} \Psi (Y)= & {} c_{Y_0,H}\left( \sum _{m=0}^{\sqrt{R}}\frac{c_{m,1}}{Y^{mc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \right) \\&- \left( \sum _{m=0}^{\sqrt{R}}\frac{c_{m,1}}{Y^{mc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \right) \\&\times \int _{Y_0}^Y \left( \sum _{m=0}^{\sqrt{R}} \frac{c_{m,2}}{\tau ^{mc_R}}+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \right) \\&\times \left( \sum _{j=1}^{\sqrt{R}} \frac{q_j}{\tau ^{1+jc_R}}+O\left( \frac{1}{\tau ^{1+(\sqrt{R}+1)c_R}}\right) \right) d\tau \\&- \left( \sum _{m=0}^{\sqrt{R}}\frac{c_{m,2}}{Y^{mc_R}} Y+O\left( \frac{1}{Y^{(\sqrt{R}+1)c_R}}\right) \right) \\&\int _Y^\infty \left( \sum _{m=0}^{\sqrt{R}}\frac{c_{m,1}}{\tau ^{mc_R}}+O\left( \frac{\mathrm{log}(\tau )}{\tau ^{(\sqrt{R}+1)c_R}}\right) \right) \\&\times \left( \sum _{j=1}^{\sqrt{R}} \frac{q_j}{\tau ^{2+jc_R}}+O\left( \frac{1}{\tau ^{2+(\sqrt{R}+1) c_R}}\right) \right) d\tau \\= & {} \sum _{m=0}^{\sqrt{R}-1} \frac{b_m}{Y^{mc_R}}+O\left( \frac{1}{Y^{\sqrt{R}c_R}}\right) . \end{aligned}$$

We therefore have proved that for \(H\in {\mathcal {C}}^\infty ([0,Z_2])\), there exists a unique solution \(\Phi \) to (3.56) on \([0,Z_2]\) which is \(o((Z_2-Z)^{-\frac{1}{c_R}})\) at \(Z=Z_2\). Furthermore, this solution is smooth on \([0,Z_2)\), and is \(C^{\sqrt{R}}\) at \(Z=Z_2\) where it admits an asymptotic expansion

$$\begin{aligned} \Phi (Z)= & {} \sum _{j=0}^{\sqrt{R} -1}c_{j,\Phi }(Z_2-Z)^j+O\Big ((Z_2-Z)^{\sqrt{R}}\Big ). \end{aligned}$$
(3.65)

Outer solution of the inhomogeneous problem. We argue similarly, considering the basis \(\Phi _1(Z)\) and \(\Phi _{rad}^+(Z)\) with \(\Phi _{rad}^+(Z_a)=1\), for \(Z_2<Z\le Z_a\) and construct \(\Phi \) solution to (3.56) on \([Z_2,Z_a]\) which is smooth on \((Z_2, Z_a]\), \(o((Z_2-Z)^{-\frac{1}{c_R}})\) at \(Z=Z_2\) and \(C^{\sqrt{R}}\) at \(Z=Z_2\). Furthermore, \(\Phi \) admits at \(Z=Z_2\) the following asymptotic expansion analogous to (3.65)

$$\begin{aligned} \Phi (Z)=\sum _{j=0}^{\sqrt{R} -1}{\tilde{c}}_{j,\Phi }(Z_2-Z)^j+O\Big ((Z_2-Z)^{\sqrt{R}}\Big ). \end{aligned}$$

The asymptotic expansion is uniquely determined from the Eq. (3.56) and the first coefficient \({\tilde{c}}_{0,\Phi }\). We now recall that the function \(\Phi _1\) belongs to \({\mathcal {C}}^{\sqrt{R}}[0,Z_a]\) and \(\Phi _1(Z_2)=1\). By adding \(\Phi _1\) to the above expansion, we obtain another solution in which we can force the condition

$$\begin{aligned} {\tilde{c}}_{0,\Phi }=c_{0,\Phi } \end{aligned}$$

with \(c_{0,\Phi }\) appearing in (3.65). As a result, the asymptotic expansions of the inner and outer solutions are matched to order \(\sqrt{R}\), so that the constructed solution is \({\mathcal {C}}^{\sqrt{R}}\) at \(Z_2\). Finally, we have shown that given any smooth function H on \([0,Z_a]\), there exists a unique solution \(\Phi \) to (3.56) on \([0,Z_a]\) which is \(o((Z_2-Z)^{-\frac{1}{c_R}})\) at \(Z=Z_2\). Furthermore, this solution is smooth for \(Z\ne Z_2\) and \(C^{\sqrt{R}}\) at \(Z=Z_2\). In particular, with T recovered by (3.54) and smooth for \(Z\ne Z_2\) and \(C^{\sqrt{R} -1}\) at \(Z=Z_2\), we have that \((\Phi , T)\in {\mathbb {H}}_{2k}\) for \(R>R(k)\) large enough. Also, since \((\Phi , T)\) with \(\Phi \sim (Z_2-Z)^{-\frac{1}{c_R}}\) near \(Z=Z_2\) does not belong to \({\mathbb {H}}_{2k}\),Footnote 11 we have now proved that, in fact, there exists a unique solution \(X=(\Phi , T)\) to \((-{\mathcal {M}}+R)X=F\) on \([0,Z_a]\) in \({\mathbb {H}}_{2k}\), which concludes the proof of (3.53).

Step 6 Density of \({\mathcal {D}}_{R}\). We now prove that \({\mathcal {D}}_{R}\) given by (3.48) is dense in \(D({\mathcal {M}})\). Indeed, if \(X\in D({\mathcal {M}})\), then \(X\in {{\mathbb {H}}}_{2k}\) and \({\mathcal {M}}X\in {{\mathbb {H}}}_{2k}\) so that there exists a sequence \((Y_n)_{n\in {{\mathbb {N}}}}\in C^\infty ([0,Z_a],{\mathbb {C}}^2)\) with

$$\begin{aligned} \mathop {\mathrm{lim}}_{n\rightarrow +\infty }Y_n\rightarrow (-{\mathcal {M}}+R)X\text { in }{{\mathbb {H}}}_{2k}. \end{aligned}$$

From step 5, for each integer n, there exist a unique \(Z_n\in {\mathcal {D}}_{R}\) solution to

$$\begin{aligned} (-{\mathcal {M}}+R)Z_n=Y_n, \ \ Z_n\in {\mathbb {H}}_{2k}, \end{aligned}$$

and hence

$$\begin{aligned} (-{\mathcal {M}}+R)Z_n\rightarrow (-{\mathcal {M}}+R)X\text { in }{{\mathbb {H}}}_{2k}. \end{aligned}$$

Thus, to conclude, it remains to check that \(Z_n\) converges to X in \({{\mathbb {H}}}_{2k}\). To this end, since \(Z_n\in {\mathcal {D}}_{R}\), (3.46) holds for \(Z_n-Z_q\) and thus:

$$\begin{aligned}&\mathfrak {R}\langle Y_n-Y_q, Z_n-Z_q\rangle = \mathfrak {R}\langle (-{\mathcal {M}}+ R )(Z_n-Z_q), Z_n-Z_q\rangle \\&\quad = \mathfrak {R}\langle (-{\mathcal {M}} +{\mathcal {A}})(Z_n-Z_q),Z_n-Z_q\rangle -\mathfrak {R}\langle {\mathcal {A}}(Z_n-Z_q),Z_n-Z_q\rangle \\&\qquad +R\Vert Z_n-Z_q\Vert _{{{\mathbb {H}}}_{2k}}^2\\&\quad \ge R\Vert Z_n-Z_q\Vert _{{{\mathbb {H}}}_{2k}}^2 -\mathfrak {R}\langle {\mathcal {A}}(Z_n-Z_q),Z_n-Z_q\rangle \end{aligned}$$

so that, since \({\mathcal {A}}\) is a bounded operator, we infer for R sufficiently large

$$\begin{aligned} \frac{R}{2}\Vert Z_n-Z_q\Vert _{{{\mathbb {H}}}_{2k}}\le & {} \Vert Y_n-Y_q\Vert _{{{\mathbb {H}}}_{2k}}. \end{aligned}$$

In view of the convergence of \((Y_n)\) in \({{\mathbb {H}}}_{2k}\), we deduce that \(Z_n\) is a Cauchy sequence in \({{\mathbb {H}}}_{2k}\) and hence converges, i.e.

$$\begin{aligned} \mathop {\mathrm{lim}}_{n\rightarrow +\infty }Z_n\rightarrow Z\text { in }{{\mathbb {H}}}_{2k}, \ \ \ \ Z\in {{\mathbb {H}}}_{2k}. \end{aligned}$$

Since \((-{\mathcal {M}}+R)Z_n\) converges to \((-{\mathcal {M}}+R)X\) in \({{\mathbb {H}}}_{2k}\), we infer

$$\begin{aligned} (-{\mathcal {M}}+R)(Z-X)=0\text { in }{\mathcal {D}}'(0,Z_a), \ \ Z-X\in {{\mathbb {H}}}_{2k}. \end{aligned}$$

The uniqueness statement in (3.53) applied for \(F=0\) yields \(Z=X\). Thus \(Z_n\rightarrow X\) and \((-{\mathcal {M}}+R)Z_n\rightarrow (-{\mathcal {M}}+R)X\) in \({{\mathbb {H}}}_{2k}\). Finally, we have obtained a sequence \(Z_n\in {\mathcal {D}}_{R}\) such that \(Z_n\rightarrow X\) in \(D({\mathcal {M}})\), and hence \({\mathcal {D}}_{R}\) is dense in \(D({\mathcal {M}})\) as claimed.

Step 7 Maximal accretivity. We have proved in steps 1 to 3 that (3.46) holds for \(X\in {\mathcal {D}}_{R}\), i.e.

$$\begin{aligned} \forall X\in {\mathcal {D}}_{R}, \ \ \mathfrak {R}\langle (-{\mathcal {M}}+{\mathcal {A}}) X,X\rangle \ge c^*ak \langle X,X\rangle . \end{aligned}$$

Since \({\mathcal {D}}_{R}\) is dense in \(D({\mathcal {M}})\), in view of step 6, we have

$$\begin{aligned} \forall X\in {\mathcal {D}}({\mathcal {M}}), \ \ \mathfrak {R}\langle (-{\mathcal {M}}+{\mathcal {A}}) X,X\rangle \ge c^*ak\langle X,X\rangle , \end{aligned}$$

which concludes the proof of the accretivity property (3.46).

We now claim:

$$\begin{aligned} \forall F\in {\mathbb {H}}_{2k}, \ \ \exists X\in D({\mathcal {M}})\ \ \text{ such } \text{ that } \ \ (-{\mathcal {M}}+ R)X=F. \end{aligned}$$
(3.66)

Indeed, since \(F\in {\mathbb {H}}_{2k}\), by density, there exists

$$\begin{aligned} \mathop {\mathrm{lim}}_{n\rightarrow +\infty }F_n\rightarrow F\text { in }{{\mathbb {H}}}_{2k}, \ \ \ \ F_n\in C^\infty ([0,Z_a]). \end{aligned}$$

Since \(F_n\in C^\infty ([0,Z_a])\), by (3.53), there exists \(X_n\in {{\mathbb {H}}}_{2k}\)—solution to

$$\begin{aligned} (-{\mathcal {M}}+ R)X_n=F_n. \end{aligned}$$

Using (3.46) and arguing as in step 6, we have for R sufficiently large

$$\begin{aligned} \frac{R}{2}\Vert X_n-X_q\Vert _{{{\mathbb {H}}}_{2k}}\le & {} \Vert F_n-F_q\Vert _{{{\mathbb {H}}}_{2k}}. \end{aligned}$$

In view of the convergence of \((F_n)\) in \({{\mathbb {H}}}_{2k}\), we deduce that \(X_n\) is a Cauchy sequence in \({{\mathbb {H}}}_{2k}\) and hence converges, i.e.

$$\begin{aligned} \mathop {\mathrm{lim}}_{n\rightarrow +\infty }X_n\rightarrow X\text { in }{{\mathbb {H}}}_{2k}, \ \ \ \ X\in {{\mathbb {H}}}_{2k}. \end{aligned}$$

On the other hand, since \((-{\mathcal {M}}+R)X_n=F_n\) converges to F in \({{\mathbb {H}}}_{2k}\), we infer

$$\begin{aligned} (-{\mathcal {M}}+ R)X=F, \ \ X\in D({\mathcal {M}}), \end{aligned}$$

which concludes the proof of (3.66).

Finally, (3.46) and a classical and elementary argumentFootnote 12 ensures that the maximality property (3.47) is implied by

$$\begin{aligned} \exists R>0, \ \ \forall F\in {\mathbb {H}}_{2k}, \ \ \exists X\in {\mathcal {D}}({\mathcal {M}})\ \ \text{ such } \text{ that } \ \ (-\tilde{{\mathcal {M}}}+R)X=F. \end{aligned}$$

Indeed, let \(R>0\) large enough and \(F\in {\mathbb {H}}_{2k}\). Since \({\mathcal {A}}\) is a bounded operator, for R large enough, from (3.66) and (3.46),

$$\begin{aligned} \mathfrak {R}\langle F,X\rangle =\mathfrak {R}\langle (-{\mathcal {M}}+ R )X,X\rangle =\mathfrak {R}\langle (-\widetilde{{\mathcal {M}}} - {\mathcal {A}}+R)X,X\rangle \ge \frac{R}{2}\Vert X\Vert _{{\mathbb {H}}_{2k}}^2. \end{aligned}$$

Therefore, for any \(F\in {\mathbb {H}}_{2k}\), solution X to (3.66) is unique. Therefore, \((-{\mathcal {M}}+ R)^{-1}\) is well defined on \({\mathbb {H}}_{2k}\) with the bound

$$\begin{aligned} \Vert (-{\mathcal {M}}+R)^{-1}\Vert _{{\mathcal {L}}({\mathbb {H}}_{2k},{\mathbb {H}}_{2k})}\lesssim \frac{1}{R}. \end{aligned}$$

Hence

$$\begin{aligned} -\widetilde{{\mathcal {M}}}+R=-{\mathcal {M}} + {\mathcal {A}}+ R=(-{\mathcal {M}}+ R)\left[ \mathrm{Id} + (-{\mathcal {M}}+ R)^{-1}{\mathcal {A}}\right] \end{aligned}$$

is invertible on \({\mathbb {H}}_{2k}\) for R large enough, which yields (3.47). This concludes the proof of Proposition 3.10. \(\square \)

4 Set up and the bootstrap

In this section we describe a set of smooth well localized initial data which lead to the conclusions of Theorem 1.1. The heart of the proof is a bootstrap argument coupled to the classical Brouwer topological argument of Lemma 3.5 to avoid finitely many unstable directions of the corresponding linear flow. Since our analysis relies essentially on the phase-modulus decomposition of solutions of the Schrödinger equation, our chosen data needs to give rise to nowhere vanishing solutions to (1.1) (at least for a sufficiently small time as in Proposition 4.1 of [9]).

4.1 Renormalized variables

Let \(u(t,x)\in {\mathcal {C}}([0,T_*),\cap _{k\ge 0} H^k)\) be a solution to (1.1) such that u(tx) does not vanish at any \((t,x)\in [0,T_*)\times {{\mathbb {R}}}^d\). This will be a consequence of our choice of initial data and suitable bootstrap assumptions. We introduce for such a solution the decomposition of Lemma 2.1

$$\begin{aligned} u(t,x)=\frac{1}{(\lambda \sqrt{b})^{\frac{2}{p-1}}}w({\tau },y)e^{i\gamma }, \ \ w(\tau ,y)=\rho _{\mathrm{Tot}}(\tau ,Z)e^{i\frac{\Psi _{\mathrm{Tot}}}{b}} \end{aligned}$$
(4.1)

with the renormalized space and times

$$\begin{aligned} \left| \begin{array}{lll} Z=y\sqrt{b}=Z^*x, \ \ Z^*=e^{\mu \tau },\\ \lambda (\tau )=e^{-\frac{\tau }{2}}, \ \ b(\tau )=e^{-{{\mathscr {e}}}\tau }, \ \ \gamma _\tau =-\frac{1}{b}=-e^{{{\mathscr {e}}}\tau },\\ \tau =-\mathrm{log}(T_*-t), \ \ \tau _0=-\mathrm{log}(T_*). \end{array}\right. \end{aligned}$$
(4.2)

Here, \(0<{{\mathscr {e}}}<1\) is the fixed front speed such that

$$\begin{aligned} r=\frac{2}{1-{{\mathscr {e}}}}>2. \end{aligned}$$

Up to a constant the phase can more explicitly be written in the form

$$\begin{aligned} \gamma (\tau )=-\frac{1}{{{\mathscr {e}}}b}. \end{aligned}$$
(4.3)

Our claim is that given

$$\begin{aligned} \tau _0=-\mathrm{log}(T_*) \end{aligned}$$

large enough, we can construct a finite co-dimensional manifold of smooth well localized initial data \(u_0\) such that the corresponding solution to the renormalized flow (2.23) is global in renormalized time \(\tau \in [\tau _0,+\infty )\), bounded in a suitable topology and nowhere vanishing. Upon unfolding (4.1), this produces a solution to (1.1) blowing up at \(T_*\) in the regime described by Theorem 1.1.

4.2 Stabilization and regularization of the profile outside the singularity

The spherically symmetric profile solution \((\rho _P,\Psi _P)\) has an intrinsic slow decay as \(Z\rightarrow +\infty \)

$$\begin{aligned} \rho _P(Z)=\frac{c_P}{{Z}^{\frac{2(r-1)}{p-1}}}\left( 1+O\left( \frac{1}{{Z}^r}\right) \right) , \end{aligned}$$

which needs to be regularized in order to produce finite energy non vanishing initial data.

1. Stabilization of the profile. Recall the asymptotics (2.20) and the choice of parameters (4.3), (4.2) which yield

$$\begin{aligned} \lambda ^{2(r-2)}=b^r, \qquad r=\frac{2}{1-{{\mathscr {e}}}},\qquad \mu =\frac{1-{{\mathscr {e}}}}{2}. \end{aligned}$$

For \(Z=\frac{\sqrt{b}}{\lambda }x\gg 1\), i.e., outside the singularity:

$$\begin{aligned} u_P(t,x)= & {} \frac{e^{i\gamma (\tau )}}{(\lambda \sqrt{b})^{\frac{2}{p-1}}}\rho _P(Z)e^{i\frac{\Psi _P}{b}}\nonumber \\= & {} \frac{c_Pe^{-\frac{i}{{{\mathscr {e}}}b}}}{(\lambda \sqrt{b})^{\frac{2}{p-1}}\left( \frac{\sqrt{b}}{\lambda }x\right) ^{\frac{2(r-1)}{p-1}}}e^{i\left[ \frac{1}{eb}+\frac{c_{\Psi }}{b\left( \frac{\sqrt{b}}{\lambda }x\right) ^{r-2}}\right] }\left( 1+O\left( \frac{1}{\langle Z\rangle ^r}\right) \right) \nonumber \\= & {} \frac{c_P}{x^{\frac{2(r-1)}{p-1}}}e^{i\frac{c_\Psi }{x^{r-2}}\left[ 1+O\left( \frac{1}{Z^r}\right) \right] }\left[ 1+O\left( \frac{1}{Z^r}\right) \right] . \end{aligned}$$
(4.4)

We see that far away from the singularity the profile \(u_P\) is stationary. It is precisely this property that will allow us to dampen the tail of the profile below and construct solutions arising from rapidly decaying (in particular, finite energy) initial data.

2. Dampening of the tail. We dampen the tail outside the singularity \(x\ge 1\), i.e., \(Z\ge Z^*\) as follows. Let

$$\begin{aligned} R_P(t,x)=\frac{1}{(\lambda \sqrt{b})^{\frac{2}{p-1}}}\rho _P(Z), \ \ x=Ze^{-\mu \tau }, \end{aligned}$$
(4.5)

then the asympotics (4.4) imply the existence of a limiting profile for \(x\ge 1\):

$$\begin{aligned} R_P(t,x)=\frac{c_P}{x^{\frac{2(r-1)}{p-1}}}\left( 1+O(e^{-\mu r\tau })\right) \end{aligned}$$

We then pick once and for all a large integer \(n_P\gg 1\) and define a smooth non decreasing connection \({\mathcal {K}}(x)\)

$$\begin{aligned} {\mathcal {K}}(x)=\left| \begin{array}{ll} 0\ \ \text{ for }\ \ |x|\le 5,\\ n_P-\frac{2(r-1)}{p-1}\ \ \text{ for }\ \ |x|\ge 10 \end{array}\right. \end{aligned}$$
(4.6)

for some large enough universal constant

$$\begin{aligned} n_P=n_P(d)\gg 1. \end{aligned}$$

We then define the dampened tail profile in original variables

$$\begin{aligned} R_D(t,x)= & {} R_P(t,x)e^{-\int _0^x\frac{{\mathcal {K}}(x')}{x'}dx'}\nonumber \\= & {} \left| \begin{array}{ll} R_P(t,x)\ \ \text{ for }\ \ |x|\le 5,\\ \frac{c_P}{x^{n_P}}\left[ 1+O\left( e^{-\mu r\tau })\right) \right] \ \ \text{ for }\ \ |x|\ge 10\end{array}\right. , \end{aligned}$$
(4.7)

and hence in renormalized variables:

$$\begin{aligned} \left| \begin{array}{l} \rho _D(\tau ,Z)=(\lambda \sqrt{b})^{\frac{2}{p-1}} R_D(t,x),\\ x=\frac{Z}{Z^*}, \ \ Z^*=e^{\mu \tau }. \end{array}\right. \end{aligned}$$
(4.8)

Let

$$\begin{aligned} \zeta (x)=e^{-\int _0^x\frac{{\mathcal {K}}(x')}{x'}dx'}, \end{aligned}$$

we have the equivalent representation:

$$\begin{aligned} \rho _D(Z)= & {} (\lambda \sqrt{b})^{\frac{2}{p-1}} R_D(\tau , x)=(\lambda \sqrt{b})^{\frac{2}{p-1}}R_P(t,x)\zeta (x)\nonumber \\= & {} \zeta \left( \frac{Z}{Z^*}\right) \rho _P(Z) \end{aligned}$$
(4.9)

Note that by construction for \(j\in {\mathbb {N}}^*\):

$$\begin{aligned} -\frac{Z^j\partial ^j_Z\rho _D}{\rho _D}=\left| \begin{array}{ll}(-1)^{j-1}\left( \frac{2(r-1)}{p-1}\right) ^j+O\left( \frac{1}{\langle Z\rangle ^r}\right) \ \ \text{ for }\ \ Z\le 5Z^*,\\ (-1)^{j-1} n_P^j+O\left( \frac{1}{\langle Z\rangle ^r}\right) \text{ for }\ \ Z\ge 10Z^* \end{array}\right. \end{aligned}$$
(4.10)

and

$$\begin{aligned} \left| \frac{Z^j\partial _j\rho _D}{\rho _D}\right| _{L^\infty }\lesssim 1. \end{aligned}$$

The obtained dampened profile for \(Z\ge Z^*\) will be denoted

$$\begin{aligned} (\rho _D,\Psi _P), \ \ Q_D=\rho _D^{p-1}. \end{aligned}$$

4.3 Initial data

We now describe explicitly an open set of initial data which will be considered as perturbations of the profile \((\rho _D,\Psi _P)\) in a suitable topology. The conclusions of Theorem 1.1 will hold for a finite co-dimension set of such data.

We pick universal constants \(0<a\ll 1\), \(Z_0\gg 1\) which will be adjusted along the proof and depend only on \((d,\ell )\). We define two levels of regularity

$$\begin{aligned} \frac{d}{2}\ll k_0\ll k_m, \end{aligned}$$

where \(k_m\) denotes the maximum level of regularity required for the solution and \(k_0\) is the level of regularity required for the linear spectral theory on the compact set \([0,Z_a]\).

0. Variables and notations for derivatives. We define the variables

$$\begin{aligned} \left| \begin{array}{l} \rho _{\mathrm{Tot}}=\rho _P+\rho =\rho _D+{\tilde{\rho }},\\ \Psi _{\mathrm{Tot}}=\Psi _P+\Psi , \\ \Phi =\rho _P\Psi , \end{array}\right. \end{aligned}$$
(4.11)

and specify the data in the \(({\tilde{\rho }},\Psi )\) variables. We will use the following notations for derivatives. Given \(k\in {\mathbb {N}}\), we note

$$\begin{aligned} \partial ^k=(\partial _1^k,\ldots ,\partial _d^k),\quad f^{(k)}:=\partial ^kf \end{aligned}$$

the vector of k-th derivatives in each direction. The notation \(\partial _Z^k f\) is the k-th radial derivative. We let

$$\begin{aligned} {\tilde{\rho }}_k=\Delta ^k {\tilde{\rho }},\quad \Psi _k=\Delta ^k\Psi . \end{aligned}$$

Given a multiindex \(\alpha =(\alpha _1,\ldots ,\alpha _d)\in {\mathbb {N}}^d\), we note

$$\begin{aligned} \partial ^\alpha =\partial ^{\alpha _1}_1\cdots \partial ^{\alpha _d}_d, \quad |\alpha |=\alpha _1+\cdots +\alpha _d. \end{aligned}$$

1. Initializing the Brouwer argument. We define the variables adapted to the spectral analysis according to (3.8), (3.12):

$$\begin{aligned} \left| \begin{array}{ll} \Phi =\rho _P\Psi , \\ T=\partial _\tau \Phi +aH_2\Lambda \Phi , \end{array}\right. \ \ X=\left| \begin{array}{ll}T\\ \Phi \end{array}\right. \end{aligned}$$
(4.12)

and recall the scalar product (3.44). For \(0<c_g,a\ll 1\) small enough, we choose \(k_0\gg 1\) such that Proposition 3.10 applies in the Hilbert space \({\mathbb {H}}_{2k_0}\) with the spectral gap

$$\begin{aligned} \forall X\in {\mathcal {D}}({\mathcal {M}}), \ \ \mathfrak {R}\langle (-{\mathcal {M}}+{\mathcal {A}}) X,X\rangle \ge c_g\langle X,X\rangle . \end{aligned}$$
(4.13)

Hence

$$\begin{aligned} {\mathcal {M}}=({\mathcal {M}}-{\mathcal {A}} +c_g)-c_g+{\mathcal {A}} \end{aligned}$$

and we may apply Lemma 3.4:

$$\begin{aligned} \Lambda _0= & {} \{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )\ge 0\} \cap \{\lambda \ \ \text{ is } \text{ an } \text{ eigenvalue } \text{ of }\ \ {\mathcal {M}}\}\nonumber \\= & {} (\lambda _i)_{1\le i\le N} \end{aligned}$$
(4.14)

is a finite set corresponding to unstable eigenvalues, V is an associated (unstable) finite dimensional invariant set, U is the complementary (stable) invariant set

$$\begin{aligned} {\mathbb {H}}_{2k_0}=U\bigoplus V \end{aligned}$$
(4.15)

and \({{\mathbb {P}}}\) is the associated projection on V. We denote by \({{\mathcal {N}}}\) the nilpotent part of the matrix, which consists of a finite collection of Jordan blocks, representing \({{\mathcal {M}}}\) on V:

$$\begin{aligned} {{\mathcal {M}}}|_V={{\mathcal {N}}} + {\text {diag}}. \end{aligned}$$
(4.16)

Note that \({{\mathcal {N}}}\) commutes with \({{\mathcal {M}}}|_V\). Then there exist \(C, \delta _g>0\) such that (3.5) holds:

$$\begin{aligned} \forall X\in U, \ \ \Vert e^{\tau {\mathcal {M}}}X\Vert _{{\mathbb {H}}_{2k_0}}\le C e^{-\frac{\delta _g}{2} \tau }\Vert X\Vert _{{\mathbb {H}}_{2k_0}},\qquad \forall \tau \ge \tau _0. \end{aligned}$$

We now choose the data at \(\tau _0\) such that (its restriction to \([0,Z_a]\), where the projection \({{\mathbb {P}}}\) and the space \({\mathbb {H}}_{2k_0}\) are defined, satisfies)

$$\begin{aligned} \Vert (I-{{\mathbb {P}}}) X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{\delta _g}{2} \tau _0},\qquad \Vert {{\mathbb {P}}}X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{3\delta _g}{5} \tau _0}. \end{aligned}$$

2. Bounds on local low Sobolev norms. Let \(0\le m\le 2k_0\) and

$$\begin{aligned} \nu _0=-\frac{2(r-1)}{p-1}+\frac{\delta _g}{2\mu }, \end{aligned}$$
(4.17)

let the weight function

$$\begin{aligned} \chi _{\nu _0,m}=\frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)}}\zeta \left( \frac{Z}{Z^*}\right) , \ \ \zeta (Z)=\left| \begin{array}{ll}1\ \ \text{ for }\ \ Z\le 2,\\ 0\ \ \text{ for }\ \ Z\ge 3. \end{array}\right. \end{aligned}$$
(4.18)

Then

$$\begin{aligned} \sum _{m=0}^{2k_0}\int (p-1)Q(\partial ^m\rho (\tau _0))^2\chi _{\nu _0,m}+|\nabla \partial ^m\Phi (\tau _0)|^2\chi _{\nu _0,m}\le e^{-\delta _g\tau _0}. \end{aligned}$$
(4.19)

4. Pointwise assumptions. We assume the following interior pointwise bounds

$$\begin{aligned} \forall 0\le & {} k\le {k_m+1}, \nonumber \\&\left\| \frac{\langle Z\rangle ^k\partial _Z^k{\tilde{\rho }}(\tau _0)}{\rho _D}\right\| _{L^\infty (Z\le Z^*)} +\Vert \langle Z\rangle ^{r-2}\langle Z\rangle ^k\partial _Z^k\Psi (\tau _0)\Vert _{L^\infty (Z\le Z^*)}\nonumber \\\le & {} b_0^{c_0} \end{aligned}$$
(4.20)

for some small enough universal constant \(c_0\), and the exterior bounds:

$$\begin{aligned} \forall 0\le & {} k\le {k_m+1}, \nonumber \\&\left\| \frac{Z^{k{+1}}\partial _Z^k{\tilde{\rho }}(\tau _0)}{\rho _D}\right\| _{L^\infty (Z\ge Z^*)} +\frac{\Vert Z^{k{+1}}\partial _Z^k\Psi (\tau _0)\Vert _{L^\infty (Z\ge Z^*)}}{b_0}\nonumber \\\le & {} b_0^{C_0} \end{aligned}$$
(4.21)

for some large enough universal \(C_0(d,r,p)\). Note in particular that (4.20), (4.21) ensure for \(0<b_0<b_0^*\ll 1\) small enough:

$$\begin{aligned} \left\| \frac{{\tilde{\rho }}(\tau _0)}{\rho _D}\right\| _{L^\infty }\le \delta _0\ll 1 \end{aligned}$$
(4.22)

and hence the data does not vanish.

5. Global rough bound for large Sobolev norms. We consider the global Sobolev norm

$$\begin{aligned}&\Vert {\tilde{\rho }},\Psi \Vert _{k_m}^2\nonumber \\&\quad :=\sum _{j=0}^{k_m}\sum _{|\alpha |=j}\int \frac{b^2|\nabla \partial ^\alpha {\tilde{\rho }}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}(\partial ^\alpha {\tilde{\rho }})^2+\rho _{\mathrm{Tot}}^2|\nabla \partial ^\alpha \Psi |^2}{\langle Z\rangle ^{2(k_m-j)}},\nonumber \\ \end{aligned}$$
(4.23)

then we require:

$$\begin{aligned} \Vert {\tilde{\rho }}(\tau _0),\Psi (\tau _0)\Vert _{k_m}\le \frac{1}{2}. \end{aligned}$$
(4.24)

The bound above is actually implied by the pointwise assumptions.

Remark 4.1

Note that we may without loss of generality assume \(u_0\in \cap _{k\ge 0}H^k.\)

4.4 Bootstrap bounds

We make the following bootstrap assumptions on the maximal interval \([\tau _0,\tau ^*)\).

0. Non vanishing and hydrodynamical variables. From standard Cauchy theory and the smoothness of the nonlinearity since \(p\in 2{\mathbb {N}}^*+1\), the smooth data \(u_0 \in \cap _{k\ge 0}H^k\) generates a unique local solution \(u\in {\mathcal {C}}([0,T_*),\cap _{k\ge 0}H^k)\) with the blow up criterion

$$\begin{aligned} T_*<+\infty \Rightarrow \mathop {\mathrm{lim}}_{t\rightarrow T_*}\Vert u(t,\cdot )\Vert _{H^{k_c}}=+\infty \end{aligned}$$
(4.25)

for some large enough \(k_c(d,p)\). To ensure non vanishing, we first note that since \(\inf _{|x|\le 10}|u_0(x)|>0\), the continuity of u in time ensures \(\inf _{|x|\le 10}|u(t,x)|>0\) for \(t\in [0,T_*]\), \(T_*>0\) small enough. For \(|x|\ge 10\), we estimate from the flow

$$\begin{aligned} \left| {r^{n_P}}{|u(t,x)|}-{r^{n_P}}{|u_0|}\right| \le \int _0^{t}{r^{n_P}}\left| \Delta u-u|u|^{p-1}\right| dt \end{aligned}$$

and hence from our choice of initial data, the non vanishing of u(tx) follows on a time interval where

$$\begin{aligned} T_*\left\| {r^{n_P}}(|\Delta u|+|u|^{p})\right\| _{L^\infty ([0,T_*),|x|\ge 10)}\le \delta \end{aligned}$$
(4.26)

for some sufficiently small universal constant \(0<\delta \ll 1\). Using spherical symmetry we can replace the above by

$$\begin{aligned}&T_*\left( \Vert \langle x\rangle ^{n_P+1-\frac{d}{2}+\epsilon }\Delta u\Vert _{L^\infty ([0,T_*); H^1)} \right. \\&\quad \left. + \Vert r^{2\epsilon } u\Vert ^{p-1}_{L^\infty ([0,T_*),|x|\ge 10)}\Vert \langle x\rangle ^{n_P+1-\frac{d}{2}-\epsilon }u\Vert _{L^\infty ([0,T_*); H^1)}\right) \le \delta \end{aligned}$$

for an arbitrarily small \(\epsilon >0\). Our initial data \(u_0\) belongs to the space

$$\begin{aligned} \cap _{k\ge 0}H^{k}\cap \Vert {\langle x\rangle ^{n_P+1-\frac{d}{2}-\epsilon }}u\Vert _{L^2}\cap \Vert {\langle x\rangle ^{n_P+3-\frac{d}{2}-\epsilon }}\Delta u\Vert _{L^2}. \end{aligned}$$

Existence of the desired time interval \([0,T_*)\) now follows from a local well-posedness for NLS in weighted Sobolev spaces which is (essentially) in [27].

We may therefore introduce the hydrodynamical variables (4.1) on such a small enough time interval and will bootstrap the smallness bound which ensures non vanishing:

$$\begin{aligned} \left\| \frac{{\tilde{\rho }}}{\rho _{\mathrm{Tot}}}\right\| _{L^\infty }\le \delta \end{aligned}$$
(4.27)

for some sufficiently small \(0<\delta =\delta (k_m)\ll 1.\)

1. Global weighted Sobolev norms. Pick a small enough universal constant \(0<{\tilde{\nu }}<{\tilde{\nu }}^*(k_m)\ll 1\), we define

$$\begin{aligned} \left| \begin{array}{l} {\nu ={\tilde{\nu }}-\frac{2(r-1)}{p-1}},\\ \sigma _\nu =\nu +\frac{d}{2}-(r-1),\\ m_0=\frac{4k_m}{9}+1 \end{array}\right. \end{aligned}$$
(4.28)

and let the continuous function:

$$\begin{aligned} \sigma (m)=\left| \begin{array}{ll} \sigma _\nu -m\ \ \text{ for }\ \ 0\le m\le m_0,\\ -\alpha (k_m-m)\ \ \text{ for }\ \ m_0\le m\le k_m\end{array}\right. \end{aligned}$$
(4.29)

with the continuity requirement at \(m_0\):

$$\begin{aligned} \alpha (k_m-m_0)=m_0-\sigma _\nu , \ \ \alpha =\frac{m_0-\sigma _\nu }{k_m-m_0}=\frac{4}{5}+O\left( \frac{1}{k_m}\right) . \end{aligned}$$
(4.30)

In particular, \(\alpha <1\). We note that for all \(1\le m\le k_m\)

$$\begin{aligned} \sigma (m-1)\ge \sigma (m)-\alpha . \end{aligned}$$
(4.31)

We also define the function

$$\begin{aligned} {{\tilde{\sigma }}}(k)= & {} \left| \begin{array}{l} n_P-\frac{2(r-1)}{p-1}-(r-2)+{2{\tilde{\nu }}}\ \ \text{ for }\ \ 0\le k\le \frac{2k_m}{3}+1,\\ \beta (k_m-k)\ \ \text{ for }\ \ \frac{2k_m}{3}+1\le k\le k_m,\end{array}\right. \nonumber \\\le & {} n_P-\frac{2(r-1)}{p-1}-(r-2)+{2{\tilde{\nu }}}, \end{aligned}$$
(4.32)

where \(\beta \) is computed through the continuity requirement at \(\frac{2k_m}{3}\):

$$\begin{aligned} \frac{k_m}{3}\beta = n_P-\frac{2(r-1)}{p-1}-(r-2)+2{\tilde{\nu }}\Leftrightarrow \beta =3\frac{n_P-\frac{2(r-1)}{p-1}-(r-2)+2{\tilde{\nu }}}{k_m}. \end{aligned}$$

We will choose \(n_P\ll k_m\), e.g. \(n_P=\frac{k_m}{30}\), so that in particular,

$$\begin{aligned} \beta <\frac{1}{10},\qquad \alpha +\beta \le 1. \end{aligned}$$

We also note that

$$\begin{aligned} {{\tilde{\sigma }}(m-1)\le {\tilde{\sigma }}(m)+\beta .} \end{aligned}$$

We then define the weighted Sobolev norm:

$$\begin{aligned} \left| \begin{array}{ll} \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma {(m)}}^2=\sum _{k=0}^m\int \chi _{m,k,\sigma {(m)}}\\ \quad \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \\ \chi _{m,k,\sigma {(m)}}(Z)=\frac{1}{\langle Z\rangle ^{2(m-k+\sigma {(m)})}} \, \xi _m\left( \frac{Z}{Z^*}\right) , \end{array}\right. , \end{aligned}$$
(4.33)

where the function

$$\begin{aligned} \xi _m(x)=\left| \begin{array}{ll} 1&{}\quad \text{ for }\quad x\le 1\\ x^{2{{\tilde{\sigma }}}(m)}&{}\quad \text{ for }\quad x>1 \end{array}\right. \end{aligned}$$

We assume the bootstrap bound:

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)}\le 1, \ \ 0\le m\le k_m-1. \end{aligned}$$
(4.34)

Remark 4.2

(Equivalence of norms) It is easy to see that the norm (4.33) is equivalent

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)}\approx & {} \sum _{k=0}^m\sum _{|\alpha |=k} \int \chi _{m,k,\sigma (m)}\nonumber \\&\left[ b^2|\nabla \nabla ^\alpha {\tilde{\rho }}|^2+\rho _D^{p-1}{|\nabla ^\alpha {\tilde{\rho }}|^2}+\rho _D^2{|\nabla \nabla ^\alpha \Psi |^2}\right] \end{aligned}$$
(4.35)

and for even m

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)}\approx & {} \sum _{k=0}^{\frac{m}{2}}\int \chi _{m,2k,\sigma (m)}\nonumber \\&\left[ b^2|\nabla {\tilde{\rho }}_k|^2+\rho _D^{p-1}{|{\tilde{\rho }}_k|^2}+\rho _D^2{|\nabla \Psi _k|^2}\right] . \end{aligned}$$
(4.36)

Let us briefly sketch the proof. First, we note that the weight function \(\chi _{m,k,\sigma }\) can be replaced by a smooth function \({{\tilde{\chi }}}_{m,k,\sigma }\) with similar properties. In particular,

$$\begin{aligned} \left| \nabla ^\alpha {{\tilde{\chi }}}_{m,k,\sigma }\right| \le C_{\alpha ,m,k,\sigma } \frac{{{\tilde{\chi }}}_{m,k,\sigma }}{\langle Z\rangle ^\alpha }. \end{aligned}$$
(4.37)

The functions \(\rho _D^2{{\tilde{\chi }}}_{m,k,\sigma }\) and \(\rho _D^{p-1}{{\tilde{\chi }}}_{m,k,\sigma }\) also obey the property above. We now consider the case \(m=2\), let \({{\hat{\chi }}}\) be a weight function obeying (4.37) and observe that

$$\begin{aligned} \int {{\hat{\chi }}} |\partial _1 \partial _2 f|^2 = \int {{\hat{\chi }}} \partial ^2_1 f \partial ^2_2 f-\int \partial _1 {{\hat{\chi }}} \partial _2 f \partial _1 \partial _2 f+ \int \partial _2 {{\hat{\chi }}} \partial _2 f \partial ^2_1 f. \end{aligned}$$

Therefore,

$$\begin{aligned} \int {{\hat{\chi }}} |\partial _1 \partial _2 f|^2\lesssim \int {{\hat{\chi }}} (|\partial ^2_1 f|^2+|\partial ^2_2 f|^2)+ \int \frac{{{\hat{\chi }}}}{\langle Z\rangle ^2} (|\partial _1 f|^2+|\partial _2 f|^2). \end{aligned}$$

Using this for \(f=\nabla {\tilde{\rho }}, {\tilde{\rho }},\nabla \Psi \) and with any mixed derivative in place of \(\partial _1\partial _2\) immediately confirms the equivalence of the norms (4.33) and (4.35) for \(m=2\). The equivalence for higher derivatives can be proved by induction. The equivalence with (4.36) follows from a similar Bochner type identity

$$\begin{aligned} \sum _{i,j=1}^d \int {{\hat{\chi }}} |\partial _i \partial _j f|^2= & {} \int {{\hat{\chi }}} |\Delta f|^2-\sum _{i,j=1}^d \int \partial _i {{\hat{\chi }}} \partial _j f \partial _i \partial _j f \\&+\sum _{i,j=1}^d \int \partial _j {{\hat{\chi }}} \partial _j f \partial _i \partial _i f \end{aligned}$$

implying

$$\begin{aligned} \sum _{i,j=1}^d \int {{\hat{\chi }}} |\partial _i \partial _j f|^2\lesssim & {} \int {{\hat{\chi }}} |\Delta f|^2+ \int \frac{{{\hat{\chi }}}}{\langle Z\rangle ^2} |\nabla f|^2\\\lesssim & {} \int {{\hat{\chi }}} |\Delta f|^2+ \int \frac{{{\hat{\chi }}}}{\langle Z\rangle ^4} |f|^2. \end{aligned}$$

This gives the equivalence of (4.33) and (4.36) for \(m=2\). Once again, higher norms follow by induction.

Finally, note that the above norm equivalences are even independent of the assumption of spherical symmetry on \({{\tilde{\rho }}}, \Psi \).

2. Global control of the highest Sobolev norm:

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{k_m}= \Vert {\tilde{\rho }},\Psi \Vert ^2_{k_m,\sigma ({m})}\le 1. \end{aligned}$$
(4.38)

3. Local decay of low Sobolev norms: for any \(0\le k\le 2k_0\), any \({1\le }{{\hat{Z}}}\le Z^*\) and universal constant \(C=C(k_0)\):

$$\begin{aligned} \Vert ({\tilde{\rho }},\Psi )\Vert _{H^k(Z\le {\hat{Z}})}\le {{\hat{Z}}}^C e^{-\frac{3\delta _g}{8}\tau }. \end{aligned}$$
(4.39)

4. Pointwise bounds:

$$\begin{aligned} \left| \begin{array}{l} \forall 0\le k\le \frac{2k_m}{3}, \ \ \Vert \frac{Z^{n(k)}\partial _Z^k{\tilde{\rho }}}{\rho _D}\Vert _{L^\infty }\le 1,\\ \forall 1\le k\le \frac{2k_m}{3},\ \ \Vert Z^{n(k)}\langle Z\rangle ^{r-2}\partial _Z^k\Psi \Vert _{L^\infty (Z \le Z^*)}+\frac{\Vert Z^{n(k)}\partial _Z^k\Psi \Vert _{L^\infty (Z\ge Z^*)}}{b}\le 1 \end{array}\right. \end{aligned}$$
(4.40)

with

$$\begin{aligned} n(k)=\left| \begin{array}{ll} k&{}\quad \text{ for }\quad k\le \frac{4k_m}{9},\\ \frac{k_m}{4}&{}\quad \text{ for }\quad \frac{4k_m}{9}<k\le \frac{2k_m}{3}. \end{array}\right. \end{aligned}$$
(4.41)

Remark 4.3

Since \(b=e^{-\mu (r-2)\tau }\), (4.20) and (4.21) imply that the initial data verify the bootstrap inequalities (4.34), (4.38), (4.40) with the bound \(e^{-c\tau _0}\) for some small universal constant c.

The heart of the proof of Theorem 1.1 is the following:

Proposition 4.4

(Bootstrap) Let \(\tau ^*\) be the maximal time with property that (see (4.16) for the definition of \({\mathcal {N}}\))

$$\begin{aligned} \Vert e^{{-\tau }{{\mathcal {N}}}} {{\mathbb {P}}}X(\tau )\Vert _{{\mathbb {H}}_{2k_0}} {<} e^{-\frac{19\delta _g}{30}\tau } \end{aligned}$$
(4.42)

for all \(\tau \in [\tau _0,\tau ^*)\) and that the bounds (4.26), (4.34), (4.38), (4.39), (4.40), (4.27) hold on \([\tau _0,\tau ^*)\) with \({\delta }^{-1},\tau _0\) large enough. Then the following holds:

1. Exit criterion. The bounds (4.26), (4.34), (4.38), (4.39), (4.40), (4.27) can be strictly improvedFootnote 13 on \([\tau _0,\tau ^*)\). Consequently, either \(\tau ^*=+\infty \) or, if \(\tau ^*<+\infty \), then

$$\begin{aligned} \Vert e^{{-\tau ^*}{{\mathcal {N}}}} {{\mathbb {P}}}X(\tau ^*)\Vert _{{\mathbb {H}}_{2k_0}} e^{\frac{19\delta _g}{30}\tau ^*}=1. \end{aligned}$$
(4.43)

2. Linear evolution. The right hand side G of the equation for \(X(\tau )\)

$$\begin{aligned} \partial _\tau X = {{\mathcal {M}}} X + G \end{aligned}$$

satisfies

$$\begin{aligned} \Vert G(\tau )\Vert _{{\mathbb {H}}_{2k_0}} \le e^{-\frac{2\delta _g}{3}\tau },\qquad \forall \tau \in [\tau _0,\tau ^*]. \end{aligned}$$
(4.44)

We will show in Sect. 8.3 that Proposition 4.4 immediately implies Theorem 1.1.

Remark 4.5

We note that the assumption (4.42) implies that

$$\begin{aligned} \Vert {{\mathbb {P}}}X(\tau )\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{\delta _g}{2}\tau }, \qquad \forall \tau \in [\tau _0,\tau ^*). \end{aligned}$$
(4.45)

We will prove the bootstrap Proposition 4.4 under the weaker assumption (4.45). Specifically, we will define \([\tau _0,\tau ^*)\) to be the maximal time interval on which (4.45) holds and will show that both the bounds (4.26), (4.34), (4.38), (4.39), (4.40), (4.27) can be improved and that G satisfies (4.44).

We now focus on the proof of Proposition 4.4 and work on a time interval \([\tau _0,\tau ^*)\), \(\tau _0<\tau ^*\le +\infty \) on which (4.26), (4.34), (4.38), (4.39), (4.40), (4.27) and (4.45) hold.

5 Control of high Sobolev norms

We first turn to the global in space control of high Sobolev norms. This is an essential step to control the b dependence of the flow and the dissipative structure which can neither be treated by spectral analysis nor perturbatively.

We claim an improvement of the bound (4.34), controlling all but the highest weighted Sobolev norm.

Proposition 5.1

There exists a universal constant \(c^*_{k_m}>0\) such that for all \(0\le m\le k_m-1\)

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma (m)}\le e^{-c^*_{k_m}\tau }. \end{aligned}$$
(5.1)

The rest of this section is devoted to the proof of Proposition 5.1. Let us outline the main steps:

  1. (1)

    First, we derive a general weighted energy identity, see (5.10), which will be used several times in the paper, and which respects the quasilinear structure of the problem. It is important that the \(b^2\Delta \) term that was neglected in the Euler approximation of the flow produces a positive term in (5.10).

  2. (2)

    Second, we show that thanks to our choice of weights, and knowing decay on the light cone, we can derive from (5.10) the differential inequality (5.13). The control of the corresponding nonlinear terms relies on classical interpolation estimates between weighted Sobolev norms.

5.1 Algebraic energy identity

We derive the energy identity for high Sobolev norms. Due to the use of the hydrodynamical variables, the identity exhibits a quasilinear structure.

Step 1 Equation for \({\tilde{\rho }}, {\Psi }\). Recall (2.23):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho _{\mathrm{Tot}}=-\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}-\frac{\mu \ell (r-1)}{2}\rho _{\mathrm{Tot}}-\left( 2\partial _Z\Psi _{\mathrm{Tot}}+\mu Z\right) \partial _Z\rho _{\mathrm{Tot}}\\ \rho _{\mathrm{Tot}}\partial _\tau \Psi _{\mathrm{Tot}}=b^2\Delta \rho _{\mathrm{Tot}}-\left[ |\nabla \Psi _{\mathrm{Tot}}|^2+\mu (r-2)\Psi _{\mathrm{Tot}}-1\right. \\ \quad \qquad \qquad \qquad \left. +\mu \Lambda \Psi _{\mathrm{Tot}}+\rho _{\mathrm{Tot}}^{p-1}\right] \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$

By construction

$$\begin{aligned} \left| \begin{array}{ll} |\nabla \Psi _P|^2+\rho _D^{p-1}+\mu (r-2)\Psi _P+\mu \Lambda \Psi _P-1=\tilde{{\mathcal {E}}}_{P,\Psi },\\ \partial _\tau \rho _D+\rho _D\left[ \Delta \Psi _P+\frac{\mu \ell (r-1)}{2}+\left( 2\partial _Z\Psi _P+\mu Z\right) \frac{\partial _Z\rho _D}{\rho _D}\right] =\tilde{{\mathcal {E}}}_{P,\rho } \end{array}\right. \end{aligned}$$
(5.2)

with \(\tilde{{\mathcal {E}}}\) supported in \(Z\ge 3Z^*\). The linearized flow is given by

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau {\tilde{\rho }}=-\rho _{\mathrm{Tot}}\Delta \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi +H_1{\tilde{\rho }}-H_2\Lambda {\tilde{\rho }}-\tilde{{\mathcal {E}}}_{P,\rho }\\ \partial _\tau \Psi =b^2\frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\left\{ H_2\Lambda \Psi +\mu (r-2)\Psi +|\nabla \Psi |^2\right. \\ \quad \qquad \left. +(p-1)\rho _D^{p-2}{\tilde{\rho }}+\text {NL}({\tilde{\rho }})\right\} -\tilde{{\mathcal {E}}}_{P,\Psi } \end{array}\right. \end{aligned}$$
(5.3)

with the nonlinear term

$$\begin{aligned} \text {NL}({\tilde{\rho }})=(\rho _D+{\tilde{\rho }})^{p-1}-\rho _D^{p-1}-(p-1)\rho _D^{p-2}{\tilde{\rho }}. \end{aligned}$$

Note that the potentials

$$\begin{aligned} H_2=\mu +2\frac{\Psi '_P}{Z}, \ \ H_1=-\left( \Delta \Psi _P+\frac{\mu \ell (r-1)}{2}\right) \end{aligned}$$

remain the same in these equations: they are not affected by the profile localization introduced by passing from \(\rho _P\) to \(\rho _D\). We recall the Emden transform formulas (2.24):

$$\begin{aligned} \left| \begin{array}{ll} H_2=\mu (1-w),\\ H_1=\frac{\mu \ell }{2}(1-w)\left[ 1+\frac{\Lambda \sigma }{\sigma }\right] ,\\ \end{array}\right. \end{aligned}$$
(5.4)

which, using (2.19), yield the bounds:

$$\begin{aligned} \left| \begin{array}{llll} H_2=\mu +O\left( \frac{1}{\langle Z\rangle ^r}\right) , \ \ H_1=-\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) ,\\ |\langle Z\rangle ^j\partial _Z^j H_1|+|\langle Z\rangle ^j\partial _Z^jH_2|\lesssim \frac{1}{\langle Z\rangle ^{r}}, \ \ j\ge 1. \end{array}\right. \end{aligned}$$
(5.5)

Our main task is now to produce an energy identity for (5.3) which respects the quasilinear nature of (5.3) and does not loose derivatives.

Step 2 Equation for derivatives. We recall the notation for the vector \(\partial ^k\):

$$\begin{aligned}\left| \begin{array}{l} \partial ^k:=(\partial _1^k,\ldots ,\partial _d^k),\\ {\tilde{\rho }}^{(k)}=\partial ^k{\tilde{\rho }}, \quad \Psi ^{(k)}=\partial ^k\Psi . \end{array}\right. \end{aligned}$$

Also, for convenience, we denote \(\partial ^1\) in various computations simply by \(\partial \).

We use

$$\begin{aligned}{}[\partial ^k,\Lambda ]=k\partial ^k \end{aligned}$$

to compute from (5.3):

$$\begin{aligned} \partial _\tau {\tilde{\rho }}^{(k)}= & {} (H_1-kH_2){\tilde{\rho }}^{(k)}-H_2\Lambda {\tilde{\rho }}^{(k)}-(\partial ^k\rho _{\mathrm{Tot}})\Delta \Psi -k\partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \nonumber \\&-\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}- 2\nabla (\partial ^k\rho _{\mathrm{Tot}})\cdot \nabla \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)} + F_1 \end{aligned}$$
(5.6)

with

$$\begin{aligned} F_1= & {} -\partial ^k\tilde{{\mathcal {E}}}_{P,\rho }+[\partial ^k,H_1]{\tilde{\rho }}-[\partial ^k,H_2]\Lambda {\tilde{\rho }}\nonumber \\&- \sum _{\left| \begin{array}{ll} j_1+j_2=k\\ j_1\ge 2, j_2\ge 1\end{array}\right. }c_{j_1,j_2}\partial ^{j_1}\rho _{\mathrm{Tot}}\partial ^{j_2}\Delta \Psi \nonumber \\&-\sum _{\left| \begin{array}{ll}j_1+j_2=k\\ j_1,j_2\ge 1\end{array}\right. }c_{j_1,j_2}\partial ^{j_1}\nabla \rho _{\mathrm{Tot}}\cdot \partial ^{j_2}\nabla \Psi . \end{aligned}$$
(5.7)

For the second equation:

$$\begin{aligned} \partial _\tau \Psi ^{(k)}= & {} b^2\left( \frac{\partial ^k\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right) \nonumber \\&-kH_2\Psi ^{(k)}-H_2\Lambda \Psi ^{(k)}-\mu (r-2)\Psi ^{(k)}-2\nabla \Psi \cdot \nabla \Psi ^{(k)} \nonumber \\&- \left[ (p-1)\rho _{D}^{p-2}{\tilde{\rho }}^{(k)}+k(p-1)(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right] +F_2\nonumber \\ \end{aligned}$$
(5.8)

with

$$\begin{aligned} F_2= & {} -\partial ^k\tilde{{\mathcal {E}}}_{P,\Psi }+b^2\left[ \partial ^k\left( \frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right) -\frac{\partial ^k\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right] \nonumber \\&- [\partial ^k,H_2]\Lambda \Psi -(p-1)\left( [\partial ^k,\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right) \nonumber \\&- \sum _{j_1+j_2=k,j_1,j_2\ge 1}\partial ^{j_1}\nabla \Psi \cdot \partial ^{j_2}\nabla \Psi -\partial ^k\text {NL}({\tilde{\rho }}). \end{aligned}$$
(5.9)

Step 3 Algebraic energy identity. Let \(\chi \) be a smooth function. We compute:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int b^2\chi |\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} \\&\quad =\frac{1}{2}\int \partial _\tau \chi \left\{ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} \\&\qquad - eb^2\int \chi |\nabla {\tilde{\rho }}^{(k)}|^2+\int \partial _\tau {\tilde{\rho }}^{(k)}\left[ -b^2\chi \Delta {\tilde{\rho }}^{(k)}\right. \\&\qquad \left. -b^2\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad + \frac{p-1}{2}\int \chi (p-2)\partial _\tau \rho _D\rho _D^{p-3}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2 \\&\qquad + \int \chi \partial _\tau \rho _{\mathrm{Tot}}\left[ \frac{p-1}{2}\rho _D^{p-2}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}|\nabla \Psi ^{(k)}|^2\right] \\&\qquad - \int \partial _\tau \Psi ^{(k)}\left[ 2\chi \rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)} \right. \\&\qquad \left. +\chi \rho _{\mathrm{Tot}}^2\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\right] . \end{aligned}$$

We compute:

$$\begin{aligned}&\int \partial _\tau {\tilde{\rho }}^{(k)}\left[ -b^2\chi \Delta {\tilde{\rho }}^{(k)}-b^2\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\quad = \int F_1\left[ -b^2\nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad + \int \left[ (H_1-kH_2){\tilde{\rho }}^{(k)}-H_2\Lambda {\tilde{\rho }}^{(k)}-(\partial ^k\rho _{\mathrm{Tot}})\Delta \Psi -2\nabla (\partial ^k\rho _{\mathrm{Tot}})\cdot \nabla \Psi \right] \\&\qquad \times \left[ -b^2\chi \Delta {\tilde{\rho }}^{(k)}-b^2\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad - \int k\partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \left[ -b^2\chi \Delta {\tilde{\rho }}^{(k)}-b^2\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\right. \\&\qquad \left. +(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] - \int (\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}) \\&\qquad \times \left[ -b^2\chi \Delta {\tilde{\rho }}^{(k)}-b^2\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\quad = b^2\int \chi \nabla F_1\cdot \nabla {\tilde{\rho }}^{(k)}+(p-1)\int \chi F_1\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\\&\qquad + \int \left[ (H_1-kH_2){\tilde{\rho }}^{(k)}-H_2\Lambda {\tilde{\rho }}^{(k)}-(\partial ^k\rho _{\mathrm{Tot}})\Delta \Psi -2\nabla (\partial ^k\rho _{\mathrm{Tot}})\cdot \nabla \Psi \right] \\&\qquad \times \left[ -b^2\nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad - \int k\partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \left[ -b^2\nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad + b^2\int \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\left( \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right) \\&\qquad - \int \chi (\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)})\\&\qquad \times \left[ -b^2\Delta {\tilde{\rho }}^{(k)}+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] . \end{aligned}$$

Similarly:

$$\begin{aligned}&- \int \partial _\tau \Psi ^{(k)}\left[ 2\chi \rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\chi \rho _{\mathrm{Tot}}^2\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\quad = -\int F_2\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\qquad - \int \left\{ b^2\left( \frac{\partial ^k\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right) \right\} \\&\qquad \times \left[ 2\chi \rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\chi \rho _{\mathrm{Tot}}^2\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\qquad - \int \Big \{-kH_2\Psi ^{(k)}-H_2\Lambda \Psi ^{(k)}-\mu (r-2)\Psi ^{(k)}-2\nabla \Psi \cdot \nabla \Psi ^{(k)}\\&\qquad - \left[ (p-1)\rho _{D}^{p-2}{\tilde{\rho }}^{(k)}+k(p-1)(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right] \Big \}\\&\qquad \times \left[ 2\chi \rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\chi \rho _{\mathrm{Tot}}^2\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\quad = \int \chi \rho ^2_T\nabla \Psi ^{(k)}\cdot \nabla F_2 - b^2\int (\partial ^k\Delta \rho _{D}+\Delta {\tilde{\rho }}^{(k)})\left[ 2\chi \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right. \\&\qquad \left. +\chi \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\qquad + b^2\int \frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\left[ 2\chi \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right. \\&\qquad \left. +\chi \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\qquad - \int \left[ -kH_2\Psi ^{(k)}-H_2\Lambda \Psi ^{(k)}-\mu (r-2)\Psi ^{(k)}-2\nabla \Psi \cdot \nabla \Psi ^{(k)}\right] \nabla \\&\qquad \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})+ \int (p-1)\rho _{D}^{p-2}{\tilde{\rho }}^{(k)}\left[ 2\chi \rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right. \\&\qquad \left. +\chi \rho _{\mathrm{Tot}}^2\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\qquad + \int k(p-1)(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\quad = \int \chi \rho ^2_T\nabla \Psi ^{(k)}\cdot \nabla F_2- b^2\int (\partial ^k\Delta \rho _{D})\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\qquad + \int (-b^2\Delta {\tilde{\rho }}^{(k)}+(p-1)\rho _D^{p-2}{\rho _{\mathrm{Tot}}}{\tilde{\rho }}^{(k)})\\&\qquad \times \left[ 2\chi \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\chi \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+\rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\right] \\&\qquad + b^2\int \frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho ^2_T}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\qquad - \int \left[ -kH_2\Psi ^{(k)}-H_2\Lambda \Psi ^{(k)}-\mu (r-2)\Psi ^{(k)}-2\nabla \Psi \cdot \nabla \Psi ^{(k)}\right] \nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\qquad + \int k(p-1)(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\nabla \\&\qquad \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)}). \end{aligned}$$

This yields the algebraic energy identity:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int b^2\chi |\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} \nonumber \\&\quad ={\frac{1}{2}\int \partial _\tau \chi \left\{ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} }\nonumber \\&\qquad - b^2\int (\partial ^k\Delta \rho _{D})\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\nonumber \\&\qquad -eb^2\int \chi |\nabla {\tilde{\rho }}^{(k)}|^2+\int \chi \frac{\partial _\tau \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\left[ \frac{p-1}{2}\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho ^2_T|\nabla \Psi ^{(k)}|^2\right] \nonumber \\&\qquad + \frac{p-1}{2}\int \chi (p-2)\frac{\partial _\tau \rho _D}{\rho _D}\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+ \int F_1\chi (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\nonumber \\&\qquad +b^2\int \chi \nabla F_1\cdot \nabla {\tilde{\rho }}^{(k)}+\int \chi \rho ^2_T\nabla F_2\cdot \nabla \Psi ^{(k)}\nonumber \\&\qquad + \int \left[ (H_1-kH_2){\tilde{\rho }}^{(k)}-H_2\Lambda {\tilde{\rho }}^{(k)}-(\partial ^k\rho _{\mathrm{Tot}})\Delta \Psi -2\nabla (\partial ^k\rho _{\mathrm{Tot}})\cdot \nabla \Psi \right] \nonumber \\&\qquad \times \left[ -b^2\nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \nonumber \\&\qquad - \int \left[ -kH_2\Psi ^{(k)}-H_2\Lambda \Psi ^{(k)}-\mu (r-2)\Psi ^{(k)}-2\nabla \Psi \cdot \nabla \Psi ^{(k)}\right] \nonumber \\&\qquad \times \nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\nonumber \\&\qquad - \int k\partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \left[ -b^2\nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+(p-1)\chi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \nonumber \\&\qquad +b^2\int \frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho ^2_T}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\nonumber \\&\qquad + \int k(p-1)(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\nonumber \\&\qquad + b^2\int \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\left( \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right) \nonumber \\&\qquad + \int (-b^2\Delta {\tilde{\rho }}^{(k)}+(p-1)\rho _{D}^{p-2}{\rho _{\mathrm{Tot}}}{\tilde{\rho }}^{(k)})\left[ \rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\right] . \end{aligned}$$
(5.10)

5.2 Weighted \(L^2\) bound for \(m\le k_m-1\)

Given \(\sigma \in {\mathbb {R}}\), we recall the notation

$$\begin{aligned} \left| \begin{array}{ll} \Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma }^2=\sum _{m=0}^k\int \chi _{k,m,\sigma }\left[ b^2|\nabla {\tilde{\rho }}_m|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_m^2\right. \\ \quad \left. +\rho _{\mathrm{Tot}}^2|\nabla \Psi _m|^2\right] ,\\ \chi _{k,m,\sigma }(Z)=\frac{1}{\langle Z\rangle ^{2(k-m+\sigma )}} {\xi _{{k}}\left( \frac{Z}{Z^*}\right) }. \end{array}\right. \end{aligned}$$

We let

$$\begin{aligned} I_{k,\sigma }= & {} \int \frac{\xi _k\left( \frac{Z}{Z^*}\right) }{\langle Z\rangle ^{2\sigma }}\left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\right. \nonumber \\&\left. +\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] . \end{aligned}$$
(5.11)

Lemma 5.2

(Weighted \(L^2\) bound) Recall the definition (4.28), (4.29) of \(\sigma (m)\) and let

$$\begin{aligned} \left| \begin{array}{l} \sigma = \sigma (k),\\ \nu +\frac{2(r-1)}{p-1}={\tilde{\nu }}, \end{array}\right. \end{aligned}$$
(5.12)

then there exists \(c_{k_m}>0\) such that for all \(0<{\tilde{\nu }}<{\tilde{\nu }}(k_m)\ll 1\) and \(b_0<b_0(k_m)\ll 1\), for all \(1\le k\le k_m-1\), \(I_k:=I_{k,\sigma (k)}\) given by (5.11) satisfies the differential inequality

$$\begin{aligned} \frac{dI_k}{d\tau }+2\mu {\tilde{\nu }}I_k\le e^{-c_{k_m}\tau }. \end{aligned}$$
(5.13)

We claim that Lemma 5.2 implies Proposition 5.1.

Proof of Proposition 5.1

Integrating (5.13) on the interval \([\tau _0,\tau ],\) with initial data prescribed at \(\tau _0\), we obtain

$$\begin{aligned} I_k(\tau ) \le e^{-2\mu {\tilde{\nu }}(\tau -\tau _0)} I_k(\tau _0) + \frac{1}{c_{k_m}-2\mu {\tilde{\nu }}}\left( e^{-2\mu {\tilde{\nu }}(\tau -\tau _0)-c_{k_m}\tau _0}-e^{-c_{k_m}\tau }\right) . \end{aligned}$$

We now recall, see Remark 4.3, that \(I_k(\tau _0)\le e^{-c\tau _0}\). Choosing \(4\mu {\tilde{\nu }}\le \min \{c,c_{k_m}\}\) we obtain that

$$\begin{aligned} I_k(\tau ) \le 2 e^{-2\mu {\tilde{\nu }}\tau }. \end{aligned}$$
(5.14)

We now recall from (4.35) and (4.36) for even m that \(\Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma }\) controls all the corresponding Sobolev norms: let a multi-index \(\alpha =(\alpha _1,\ldots ,\alpha _d)\) with

$$\begin{aligned} \alpha _1+\dots +\alpha _d=|\alpha |,\quad \nabla ^\alpha :=\partial _1^{\alpha _1}\cdots \partial ^{\alpha _d}_d, \end{aligned}$$

then for all \(|\alpha |=k, \quad 0\le k\le m\),

$$\begin{aligned}&b^2\int \chi _{k,m,\sigma }|\nabla \nabla ^\alpha {\tilde{\rho }}|^2+(p-1)\int \chi _{k,m,\sigma }\rho _D^{p-2}\rho _{\mathrm{Tot}}|\nabla ^\alpha {\tilde{\rho }}|^2\nonumber \\&\qquad +\int \chi _{k,m.\sigma }\rho _{\mathrm{Tot}}^2|\nabla \nabla ^\alpha \Psi |^2, \nonumber \\&\quad \lesssim \Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma }^2, \end{aligned}$$
(5.15)

and similarly the norm \(\Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma }^2\) (with even k) is equivalent to the one where \(\partial ^m\) with \(1\le m\le k\) derivatives are replaced by \(\Delta ^m\) with \(1\le m \le \frac{k}{2}\).

We now claim

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma (m)}^2\le \sum _{k=0}^m I_{k,\sigma (k)}. \end{aligned}$$
(5.16)

Combining this with (5.14) concludes the proof of (5.1) (with \(c^*_{k_m}=\mu {\tilde{\nu }}\)).

Proof of (5.16). Indeed,

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma (m)}^2= & {} \sum _{k=0}^m\int \chi _{m,k,\sigma (m)}\\&\quad \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \\= & {} \sum _{k=0}^m\int \frac{\langle Z\rangle ^{2k}}{\langle Z\rangle ^{2(m+\sigma (m))}}\xi _m(x)\\&\quad \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2 +\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \end{aligned}$$

and

$$\begin{aligned} \sum _{k=0}^m I_{k,\sigma (k)}= & {} \sum _{k=0}^m\int \frac{\xi _k(x)}{\langle Z\rangle ^{2\sigma (k)}}\\&\quad \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \\= & {} \sum _{k=0}^m\int \frac{\langle Z\rangle ^{2k}\xi _k(x)}{\langle Z\rangle ^{2(\sigma (k)+k)}}\\&\quad \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \end{aligned}$$

and hence (5.16) follows from \(\sigma (k)+k\le \sigma (m)+m\) and \(\xi _k(x)\ge \xi _m(x)\) for \(0\le k\le m\). \(\square \)

5.3 Proof of Lemma 5.2

This follows from the energy identity (5.10) coupled with the pointwise bound (4.40) to control the nonlinear term.

Step 1 Interpolation bounds. In what follows we use the convention \(\lesssim \) to denote any dependence on the universal constants, including \(k_m\). Constants \(c, c_{k_m}\) will stand for generic, universal small constants.

Our main technical tool below will be the following interpolation bound: for any \(0\le m\le k_m-1\) and \(\delta >0\), there exists \(c_{\delta ,k_m}>0\) such that

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)+\delta }\le e^{-c_{\delta ,k_m}\tau }. \end{aligned}$$
(5.17)

Indeed, the claim follows by interpolating the local decay bootstrap bound (4.39) and the bound (4.38) for the highest Sobolev norm for \(Z\le Z^*_c:=(Z^*)^c\) and using the global weighted Sobolev bound for (4.34) for \(Z\ge Z^*_c\)

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)+\delta }\le & {} (Z_{{c}}^*)^{C_{k_m}}e^{-c_{k_m}\tau }+\frac{1}{(Z^*_c)^{2\delta }}\Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)}\nonumber \\\le & {} e^{-c_{\delta ,k_m}\tau } \end{aligned}$$
(5.18)

Above, on the set \(Z\le Z^*_c\), we can replace the norm \(\Vert {\tilde{\rho }},\Psi \Vert ^2_{m,\sigma (m)+\delta }\) by \((Z_{{c}}^*)^{C_{k_m}}\) with some large constant \(C_{k_m}\), times the unweighted Sobolev norm \(\Vert {\tilde{\rho }},\Psi \Vert ^2_{H^m(Z\le Z_c^*)}\) and then interpolate the latter between the Sobolev bounds (4.39) and (4.38). That will bring an additional factor \((Z_c^*)^C\), which can be absorbed by \(C_{k_m}\), and the decaying factor \(e^{-c_{k_m}\tau }\) with a small constant \(c_{k_m}\), explicitly dependent on \(k_m\) and \(\delta \). We can then choose c small enough (dependent on \(C_{k_m}\)) to obtain the second inequality in (5.18).

We will also use the bound for the damped profile from (4.7), (4.8) and (4.9):

$$\begin{aligned} |Z^k\partial _Z^k\rho _D|\lesssim \frac{1}{\langle Z\rangle ^{\frac{2(r-1)}{p-1}}}{} \mathbf{1}_{Z\le Z^*}+\frac{1}{(Z^*)^{\frac{2(r-1)}{p-1}}}\frac{1}{\left( \frac{Z}{Z^*}\right) ^{n_P}}\mathbf{1}_{Z\ge Z^*}. \end{aligned}$$
(5.19)

We will also use the bound

$$\begin{aligned} \chi _{k-1,k-1,\sigma (k-1)}\le \langle Z\rangle ^{2(\alpha +\beta )}\chi _{k,k,\sigma (k)}\le \langle Z\rangle ^{2}\chi _{k,k,\sigma (k)}, \end{aligned}$$
(5.20)

which follows from

$$\begin{aligned} \sigma (k-1)+\alpha \ge \sigma (k),\qquad {{\tilde{\sigma }}}(k-1)\le {{\tilde{\sigma }}}(k)+\beta \end{aligned}$$
(5.21)

and \(\alpha +\beta \le 1\).

Step 2 Energy identity. We run (5.10) with

$$\begin{aligned} \chi =\frac{1}{\langle Z\rangle ^{2\sigma }} \xi _k\left( \frac{Z}{Z^*}\right) , \ \ \sigma =\sigma (k), \ \ 1\le k\le k_m-1 \end{aligned}$$
(5.22)

with \(\xi _k(x)=1\) for \(x\le 1\) and \(\xi _k(x)=x^{2{{\tilde{\sigma }}}(k)}\) for \(x>1\), and estimate all terms. In our notations

$$\begin{aligned} \chi =\chi _{k,k,\sigma (k)}. \end{aligned}$$

From (4.28), (4.29) and recalling \(m_0=\frac{4k_m}{9}+1\):

$$\begin{aligned} \sigma (k)+k= & {} \left| \begin{array}{l}\sigma _\nu \ \ \text{ for }\ \ 0\le k\le m_0\\ -\alpha (k_m-k)+k=(\alpha +1)(k-m_0)+\sigma _\nu \ \ \text{ for }\ \ m_0\le k\le k_m\end{array}\right. \nonumber \\\ge & {} \sigma _\nu \end{aligned}$$
(5.23)

and

$$\begin{aligned} {{\tilde{\sigma }}}(k)= & {} \left| \begin{array}{l} n_P-\frac{2(r-1)}{p-1}-(r-2)+2{\tilde{\nu }}\ \ \text{ for }\ \ 0\le k\le \frac{2k_m}{3}+1,\\ \beta (k_m-k)\ \ \text{ for }\ \ \frac{2k_m}{3}+1\le k\le k_m,\end{array}\right. \nonumber \\\le & {} n_P-\frac{2(r-1)}{p-1}-(r-2)+2{\tilde{\nu }}, \end{aligned}$$
(5.24)

which implies

$$\begin{aligned} \chi= & {} \frac{1}{\langle Z\rangle ^{2\sigma (k)}}\xi _k\left( \frac{Z}{Z^*}\right) \nonumber \\\lesssim & {} \frac{1}{\langle Z\rangle ^{2\sigma (k)}} \left[ 1+ \left( \frac{Z}{Z^*}\right) ^{ 2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}\mathbf{1}_{Z\ge Z^*}\right] \nonumber \\\lesssim & {} \frac{1}{\langle Z\rangle ^{-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-\frac{2(r-1)}{p-1}-(r-1)\right) }} +\frac{\left( \frac{Z}{Z^*}\right) ^{ 2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}}{\langle Z\rangle ^{-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-\frac{2(r-1)}{p-1}-(r-1)\right) }}\mathbf{1}_{Z\ge Z^*}.\nonumber \\ \end{aligned}$$
(5.25)

which we will use below. The following additional inequality will be of particular significance (\(b=(Z^*)^{2-r}\)):

$$\begin{aligned} \rho _D^2 \chi\lesssim & {} \frac{1}{\langle Z\rangle ^{-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-(r-1)\right) }} \left[ \mathbf{1}_{Z\le Z^*}+ {\left( \frac{Z}{Z^*}\right) ^{4{\tilde{\nu }}-2(r-2)}} \mathbf{1}_{Z\ge Z^*}\right] \nonumber \\= & {} \frac{1}{\langle Z\rangle ^{-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-(r-1)\right) }} \mathbf{1}_{Z\le Z^*}+ \frac{1}{b^{2-\frac{4{\tilde{\nu }}}{r-2}} \langle Z\rangle ^{-2k+2\left( \frac{d}{2}-{\tilde{\nu }}-1\right) }} \mathbf{1}_{Z\ge Z^*}\nonumber \\ \end{aligned}$$
(5.26)

Step 3 Leading order terms. In what follows, we will systematically use the standard Pohozhaev identity:

$$\begin{aligned} \int \Delta g F\cdot \nabla gdx= & {} \sum _{i,j=1}^d \int \partial _i^2 g F_j\partial _jgdx=-\sum _{i,j=1}^d \int \partial _ig(\partial _iF_j\partial _jg+F_j\partial ^2_{i,j}g)\nonumber \\= & {} -\sum _{i,j=1}^d \int \partial _iF_j\partial _ig\partial _jg+\frac{1}{2}\int |\nabla g|^2\nabla \cdot F \end{aligned}$$
(5.27)

which becomes in the case of spherically symmetric functions

$$\begin{aligned} \int _{{\mathbb {R}}^d} f\Delta g \partial _rgdx= & {} c_d\int _{{\mathbb {R}}^+}\frac{f}{r^{d-1}} \partial _r(r^{d-1}\partial _rg)r^{d-1}\partial _rg\,dr \\= & {} -\frac{1}{2}\int _{{\mathbb {R}}^d}|\partial _rg|^2\left[ f'-\frac{d-1}{r}f\right] dx, \end{aligned}$$

Cross terms. We consider

$$\begin{aligned} A_1=b^2k\left[ \int \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+\frac{\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho ^2_T}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\right] . \end{aligned}$$

We compute

$$\begin{aligned}&\partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \nabla \cdot (\chi \nabla {\tilde{\rho }}^{(k)})+\frac{\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho ^2_T}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\\&\quad = \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \left[ \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+\chi \Delta {\tilde{\rho }}^{(k)}\right] \\&\qquad + \partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\left[ \nabla \chi \cdot \nabla \Psi ^{(k)} \right. \\&\qquad \left. +2\chi \frac{\nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\cdot \nabla \Psi ^{(k)}+\chi \Delta \Psi ^{(k)}\right] \\&\quad = \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\\&\qquad +2\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\chi \frac{\nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\cdot \nabla \Psi ^{(k)}\\&\qquad + \chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \Delta {\tilde{\rho }}^{(k)}+\chi \partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}. \end{aligned}$$

The last 2 terms require an integration by parts:

$$\begin{aligned}&b^2k\left| \int \left[ \chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \Delta {\tilde{\rho }}^{(k)}+\chi \partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right] \right| \\&\quad = b^2k\left| \int \left[ -(\Delta \partial ^{k-1}{\tilde{\rho }})\partial (\chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi )+\chi \partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right] \right| \\&\quad = b^2k\left| \int \left[ -\partial ^{k-1}\Delta {\tilde{\rho }}\left[ \chi \partial ^2\rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi +\partial \chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \right] \right. \right. \\&\qquad \left. \left. +\chi \partial ^{k-1}\Delta \rho _D\partial \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right] \right| \\&\quad \lesssim C_kb^2\int \chi |\partial ^{k-1}\Delta \Psi |\left[ |\partial ^{k-1}\Delta {\tilde{\rho }}\partial ^2\rho _{\mathrm{Tot}}|\right. \\&\qquad \left. +\frac{|\partial ^{k-1}\Delta {\tilde{\rho }}\partial \rho _{\mathrm{Tot}}|}{\langle Z\rangle }+|\partial (\partial ^{k-1}\Delta \rho _D\partial \rho _{\mathrm{Tot}})|\right] \\&\quad \lesssim C_kb^2\int \chi \rho _{\mathrm{Tot}}|\partial ^{k-1}\Delta \Psi |\left[ \frac{\rho _D}{\langle Z\rangle ^{k+2}}+\frac{|\partial ^{k-1}\Delta {\tilde{\rho }}|}{\langle Z\rangle }\right] \\&\quad \lesssim \sum _{|\alpha |=k}\int \frac{\chi \rho _{\mathrm{Tot}}^2}{\langle Z\rangle }|\nabla \partial ^\alpha \Psi |^2+c_kb^4\int \frac{\chi }{\langle Z\rangle }|\nabla \partial ^\alpha {\tilde{\rho }}|^2+b^4\int \chi \frac{\rho _{D}^2}{\langle Z\rangle ^{2k+3}}, \end{aligned}$$

where in penultimate inequality we used the pointwise bound (4.40).

We now estimate the source term from ():

$$\begin{aligned} b^4\int \chi \frac{\rho _D^2}{\langle Z\rangle ^{2k+3}}\lesssim & {} b^4\int _{Z\le Z^*}\frac{Z^{d-1}dZ}{\langle Z \rangle ^{2k+3-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-(r-1)\right) }}\nonumber \\&\quad + b^2\int _{Z\ge Z^*}\left( \frac{Z}{Z^*}\right) ^{4{\tilde{\nu }}}\frac{Z^{d-1}dZ}{\langle Z \rangle ^{2k+3-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-1\right) }}\nonumber \\\lesssim & {} b^4\int _{Z\le Z^*}\langle Z\rangle ^{2(r-2)-2-2{\tilde{\nu }}}dZ\nonumber \\&\quad +b^2\int _{Z\ge Z^*}\left( \frac{Z}{Z^*}\right) ^{4{\tilde{\nu }}}{\langle Z\rangle ^{-2-2{\tilde{\nu }}}} dZ\nonumber \\\lesssim & {} b^4(Z^*)^{2(r-2)-1-2{\tilde{\nu }}}\lesssim e^{-c\tau } \end{aligned}$$
(5.28)

and hence, using (5.18),

$$\begin{aligned}&b^2k\left| \int \left[ \chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \Delta {\tilde{\rho }}^{(k)}+\chi \partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right] \right| \nonumber \\&\quad \lesssim e^{-c\tau }+\Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma +\frac{1}{2}} \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.29)

We estimate similarly,

$$\begin{aligned}&kb^2\left| \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}+\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)} \right. \nonumber \\&\qquad \left. +2\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}\chi \frac{\nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\cdot \nabla \Psi ^{(k)}\right| \nonumber \\&\quad \lesssim \sum _{|\alpha |=k}\left[ b^4\int \frac{\chi }{\langle Z\rangle }|\nabla \partial ^\alpha {\tilde{\rho }}|^2+\int \frac{\chi }{\langle Z\rangle ^{3}}\rho _{\mathrm{Tot}}^2|\nabla \partial ^\alpha \Psi |^2\right] +b^4\int \chi \frac{\rho _{D}^2}{\langle Z\rangle ^{2k+3}}\nonumber \\&\quad \lesssim e^{-c_{k_m}\tau }+\Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma +\frac{1}{2}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.30)

The remaining cross terms are estimated as follows.

$$\begin{aligned}&k(p-1)\left| \int \chi \partial \rho _{\mathrm{Tot}}\partial ^{k-1}\Delta \Psi \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right| \nonumber \\&\quad \lesssim c_k\int \chi \frac{\rho _{\mathrm{Tot}}^{p-1}}{\langle Z\rangle }\rho _{\mathrm{Tot}}|\partial ^{k-1}\Delta \Psi ||{\tilde{\rho }}^{(k)}|\nonumber \\&\quad \lesssim \int \frac{\chi }{\langle Z\rangle }\rho _D^{p-1}({\tilde{\rho }}^{(k)})^2+\int \frac{\chi }{\langle Z\rangle }\rho _{\mathrm{Tot}}^2|\nabla \partial ^\alpha \Psi |^2\nonumber \\&\quad \le \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma +\frac{1}{2}}\lesssim e^{-c_{k_m}\tau }, \end{aligned}$$
(5.31)

where we used that \(p\ge 1\) and a trivial bound \(|\rho _D|\lesssim 1\). Similarly,

$$\begin{aligned}&\int \left| (p-1)\rho _D^{p-2}{\tilde{\rho }}^{(k)}\rho ^{2}_T\nabla \chi \cdot \nabla \Psi ^{(k)}\right| \nonumber \\&\quad \lesssim \int \frac{\chi }{\langle Z\rangle }\rho _D^{p-1}({\tilde{\rho }}^{(k)})^2+\int \frac{\chi }{\langle Z\rangle }\rho _{\mathrm{Tot}}^2|\nabla \partial ^\alpha \Psi |^2\nonumber \\&\quad \le \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma +\frac{1}{2}} \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.32)

The other remaining cross term is estimated using an integration by parts:

$$\begin{aligned}&k(p-1)(p-2)\left| \int \nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right| \nonumber \\&\quad \lesssim \int \frac{\chi }{\langle Z\rangle }\rho _D^{p-1}|\nabla {\tilde{\rho }}_{k-1}|^2+ \int \frac{\chi }{\langle Z\rangle ^3}\rho _D^{p-1}{\tilde{\rho }}_{k-1}^2+\int \frac{\chi }{\langle Z\rangle }\rho _{\mathrm{Tot}}^2|\nabla \partial ^\alpha \Psi |^2\nonumber \\&\quad \le \Vert \rho ,\Psi \Vert _{k,\sigma +\frac{1}{2}}^2 \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.33)

\(\rho _k\) terms. We compute using (5.5):

$$\begin{aligned}&\int \chi (H_1-kH_2){\tilde{\rho }}^{(k)}(-b^2\Delta {\tilde{\rho }}^{(k)}+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)})\\&\qquad -b^2\int \left[ H_1-kH_2\right] {\tilde{\rho }}^{(k)}\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)} \\&\quad = \int \chi (H_1-kH_2)\left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right] \\&\qquad - \frac{b^2}{2}\int ({\tilde{\rho }}^{(k)})^2\nabla \cdot \left[ \chi \nabla (H_1-kH_2)\right] \\&\quad = -\int \mu \chi \left( k+\frac{2(r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^{r}}\right) \right) \\&\qquad \times \left( b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right) \\&\qquad - \frac{b^2}{2}\int ({\tilde{\rho }}^{(k)})^2\nabla \cdot \left[ \chi \nabla (H_1-kH_2)\right] \\&\quad = {O(e^{-c_{k_m}\tau })} -\int \mu \chi \left( k+\frac{2(r-1)}{p-1}\right) \\&\qquad \times \left( b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right) \\&\qquad - \frac{b^2}{2}\int \chi ({\tilde{\rho }}^{(k)})^2\left[ \frac{\Lambda \chi \Lambda (H_1-kH_2)}{\chi Z^2}+\Delta (H_1-kH_2) \right] , \end{aligned}$$

where we used the interpolation bound (5.18). Similarly, using that \(\chi _{k,k,\sigma }=\langle Z\rangle ^2 \chi _{k,k-1,\sigma }\) and \(|\rho _k|\le |\nabla \rho _{k-1}|\) as well as (5.5), (5.18) gives

$$\begin{aligned}&\frac{b^2}{2}\int \chi ({\tilde{\rho }}^{(k)})^2\left[ \frac{\Lambda \chi \Lambda (H_1-kH_2)}{\chi Z^2}+\Delta (H_1-kH_2) \right] \nonumber \\&\quad \lesssim \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma (k)+\frac{1}{2}(1+r)} \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.34)

Next using

$$\begin{aligned} |\partial ^k\rho _D|\lesssim \frac{\rho _D}{\langle Z\rangle ^{k}}, \end{aligned}$$

we estimate after an integration by parts:

$$\begin{aligned}&b^2\left| \int (\chi \Delta {\tilde{\rho }}^{(k)}+\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)})\left[ (\partial ^k\rho _D)\Delta \Psi +2\nabla (\partial ^k\rho _D)\cdot \nabla \Psi \right] \right| \\&\quad \lesssim b^2\int \chi |\nabla {\tilde{\rho }}^{(k)}|\left[ |\nabla (\partial ^k\rho _D\Delta \Psi )|+|\nabla (\nabla \partial ^k\rho _D\cdot \nabla \Psi )|\right. \\&\qquad \left. +\frac{|(\partial ^k\rho _D)\Delta \Psi +2\nabla (\partial ^k\rho _D)\cdot \nabla \Psi |}{\langle Z\rangle }\right] \\&\quad \le b^2\int \chi \frac{|\nabla {\tilde{\rho }}^{(k)}|^2}{\langle Z\rangle }+b^2\sum _{j=1}^3\int \chi \frac{\langle Z\rangle \rho _D^2}{\langle Z\rangle ^{2k}}\left( \frac{|\partial ^j\Psi |}{\langle Z\rangle ^{3-j}}\right) ^2. \end{aligned}$$

We use the pointwise bootstrap bound (4.40)

$$\begin{aligned} |\langle Z\rangle ^j\partial ^j\Psi |\le C_K\left[ \frac{\mathbf{1}_{Z\le Z^*}}{\langle Z\rangle ^{r-2}}+b\right] \lesssim \left[ \frac{\mathbf{1}_{Z\le Z^*}}{\langle Z\rangle ^{r-2}}+\frac{\mathbf{1}_{Z\ge Z^*}}{\langle Z^*\rangle ^{r-2}}\right] , \ \ 1\le j\le 3\nonumber \\ \end{aligned}$$
(5.35)

to estimate from ():

$$\begin{aligned}&b^2\sum _{j=1}^3\int \chi \frac{\langle Z\rangle \rho _D^2}{\langle Z\rangle ^{2k}}\left( \frac{|\partial ^j\Psi |}{\langle Z\rangle ^{3-j}}\right) ^2\\&\quad \le b^2C_K\int \frac{Z^{d-1}dZ}{\langle Z\rangle ^{2({\tilde{\nu }}+\frac{d}{2}-(r-1))}}\frac{1}{\langle Z\rangle ^{5}}\left( \left[ \frac{\mathbf{1}_{Z\le Z^*}}{\langle Z\rangle ^{2(r-2)}}+\left( \frac{Z}{Z^*}\right) ^{4{\tilde{\nu }}}\frac{\mathbf{1}_{Z\ge Z^*}}{\langle Z\rangle ^{2(r-2)}}\right] \right) \\&\quad \lesssim e^{-c_{k_m}\tau } \end{aligned}$$

and hence

$$\begin{aligned} b^2\left| \int (\chi \Delta {\tilde{\rho }}^{(k)}+\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)})\left[ (\partial ^k\rho _D)\Delta \Psi +2\nabla (\partial ^k\rho _D)\cdot \nabla \Psi \right] \right| \lesssim e^{-c_{k_m}\tau }.\nonumber \\ \end{aligned}$$
(5.36)

For the nonlinear term, we use the Pohozhaev identity (5.27) and the pointwise bound (5.35)

$$\begin{aligned} |Z^j \partial ^j \Psi |\lesssim \langle Z\rangle ^{-(r-2)}, \quad j=1,\ldots ,3 \end{aligned}$$

to estimate by the interpolation bound (5.18)

$$\begin{aligned}&b^2\left| \int (\chi \Delta {\tilde{\rho }}^{(k)}+\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)})\left[ {\tilde{\rho }}^{(k)}\Delta \Psi +2\nabla {\tilde{\rho }}^{(k)}\cdot \nabla \Psi \right] \right| \nonumber \\&\quad \lesssim b^2\left[ \int \chi \frac{|\nabla {\tilde{\rho }}^{(k)}|^2}{\langle Z\rangle ^{r}}+\int \chi \frac{({\tilde{\rho }}^{(k)})^2}{\langle Z\rangle ^{r+2}}\right] \nonumber \\&\quad \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.37)

Note that the last term in the case \(k=0\) should be treated with the help of the bound \({\tilde{\rho }}\lesssim \rho _D\) and the estimate (5.28). For \(k\ne 0\), we simply use \(|\rho _k|\le |\nabla \rho _{k-1}|\). We recall that by definition of the norm:

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma }^2\gtrsim \sum _{m=0}^k\int \chi _{k,k,\sigma }\frac{\rho _{\mathrm{Tot}}^2|\partial ^m\nabla \Psi |^2}{\langle Z\rangle ^{2(k-m)}}\gtrsim \sum _{m=1}^{k+1}\int \chi \frac{\rho _{\mathrm{Tot}}^2|\partial ^m\Psi |^2}{\langle Z\rangle ^{2(k+1-m)}}. \end{aligned}$$

Hence, by the interpolation bound,

$$\begin{aligned}&\left| \int \chi \left[ (\partial ^k\rho _D)\Delta \Psi +2\nabla (\partial ^k\rho _D)\cdot \nabla \Psi \right] (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right| \nonumber \\&\quad \lesssim \int \chi \frac{\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2}{\langle Z\rangle }+\int \chi \rho _{\mathrm{Tot}}^{p-2}\rho _{\mathrm{Tot}}^2\left[ \frac{|\partial ^2\Psi |^2}{\langle Z\rangle ^{2k-1}}+\frac{|\partial \Psi |^2}{\langle Z\rangle ^{2(k+1)-1}}\right] \nonumber \\&\quad \le \Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma +\frac{1}{2}}^2\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.38)

For the nonlinear term, we integrate by parts and use (5.35):

$$\begin{aligned}&\left| \int \chi \left[ {\tilde{\rho }}^{(k)}\Delta \Psi +2\nabla {\tilde{\rho }}^{(k)}\cdot \nabla \Psi \right] (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right| \nonumber \\&\quad \lesssim \int \chi \frac{\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2}{\langle Z\rangle }\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.39)

From Pohozhaev (5.27) and (5.5):

$$\begin{aligned}&-\int H_2\chi \Lambda {\tilde{\rho }}^{(k)}(-b^2\Delta {\tilde{\rho }}^{(k)})+b^2\int H_2\Lambda \rho _k\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)} \nonumber \\&\quad = b^2\left[ \int \Delta {\tilde{\rho }}^{(k)}(Z\chi H_2)\cdot \nabla {\tilde{\rho }}^{(k)}+\int H_2\Lambda \rho _k\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\right] \nonumber \\&\quad = b^2\left[ -\sum _{i,j=1}^d\int \partial _i(Z_j\chi H_2)\partial _i{\tilde{\rho }}^{(k)}\partial _j{\tilde{\rho }}^{(k)}+\frac{1}{2}\int |\nabla {\tilde{\rho }}^{(k)}|^2\nabla \cdot (Z\chi H_2) \right. \nonumber \\&\qquad \left. +\sum _{i,j=1}^d\int H_2Z_j\partial _j{\tilde{\rho }}^{(k)}\partial _i\chi \partial _i {\tilde{\rho }}^{(k)}\right] \nonumber \\&\quad = b^2\Bigg \{-\sum _{i,j=1}^d\int \partial _i{\tilde{\rho }}^{(k)}\partial _j{\tilde{\rho }}^{(k)}\left[ \delta _{ij}\chi H_2+Z_j\partial _i\chi H_2\right. \nonumber \\&\qquad \left. +Z_j\chi \partial _iH_2-H_2Z_j\partial _i\chi \right] \nonumber \\&\qquad +\frac{1}{2}\int |\nabla {\tilde{\rho }}^{(k)}|^2\chi H_2\left[ d+\frac{\Lambda \chi }{\chi }+\frac{\Lambda H_2}{H_2}\right] \Bigg \} \nonumber \\&\quad = \frac{\mu }{2}b^2\int \chi |\nabla {\tilde{\rho }}^{(k)}|^2\left[ d-2+\frac{\Lambda \chi }{\chi }+O\left( \frac{1}{\langle Z\rangle ^{r-1}}\right) \right] . \end{aligned}$$
(5.40)

Integrating by parts and using (5.5):

$$\begin{aligned}&-\int \chi H_2\Lambda {\tilde{\rho }}^{(k)}\left[ (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \\&\qquad +\frac{p-1}{2}\int \chi (p-2)\partial _\tau \rho _D\rho _D^{p-3}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\\&\qquad + \frac{p-1}{2}\int \chi \partial _\tau \rho _{\mathrm{Tot}}\rho _D^{p-2}({\tilde{\rho }}^{(k)})^2\\&\quad = \frac{p-1}{2}\int ({\tilde{\rho }}^{(k)})^2\left[ \nabla \cdot (Z\chi H_2\rho _D^{p-2}\rho _{\mathrm{Tot}})\right. \\&\qquad \left. +\chi \rho _{\mathrm{Tot}}\partial _\tau (\rho _D^{p-2})+\chi \partial _\tau \rho _{\mathrm{Tot}}\rho _D^{p-2}\right] \\&\quad = \frac{p-1}{2}\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\Bigg [\mu d+\mu \frac{\Lambda \chi }{\chi }+(p-2)\left( \frac{\partial _\tau \rho _D+\mu \Lambda \rho _D}{\rho _D}\right) \\&\qquad +\frac{\partial _\tau \rho _{\mathrm{Tot}}+\mu \Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}} +O\left( \frac{1}{\langle Z\rangle ^{r-1}}\right) \Bigg ]. \end{aligned}$$

We now claim the fundamental behavior

$$\begin{aligned} \frac{\partial _\tau \rho _D+\mu \Lambda \rho _D}{\rho _D}=-\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) \end{aligned}$$
(5.41)

and

$$\begin{aligned} \frac{\partial _\tau \rho _{\mathrm{Tot}}+\mu \Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}=-\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) . \end{aligned}$$
(5.42)

Assume (5.41), (5.42), we obtain

$$\begin{aligned}&-\int \chi H_2\Lambda {\tilde{\rho }}^{(k)}\left[ (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}\right] \nonumber \\&\qquad +\frac{p-1}{2}\int \chi (p-2)\partial _\tau \rho _D\rho _D^{p-3}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\nonumber \\&\qquad + \frac{p-1}{2}\int \partial _\tau \rho _{\mathrm{Tot}}{\tilde{\rho }}^{p-2}({\tilde{\rho }}^{(k)})^2\nonumber \\&\quad = \mu \frac{p-1}{2}\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\left[ d+\frac{\Lambda \chi }{\chi }-2(r-1)+O\left( \frac{1}{\langle Z\rangle ^r}\right) \right] \nonumber \\&\quad = \mu \frac{p-1}{2}\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2\left[ d+\frac{\Lambda \chi }{\chi }-2(r-1)\right] \nonumber \\&\qquad +O\left( e^{-c_{k_m}\tau }\right) . \end{aligned}$$
(5.43)

Proof of (5.41). From (4.9):

$$\begin{aligned}&\partial _\tau \rho _D+\mu \Lambda \rho _D=-\mu \Lambda \zeta \left( \frac{Z}{Z^*}\right) \rho _P(Z)+\mu \Lambda \zeta \left( \frac{Z}{Z^*}\right) \rho _P(Z), \\&\quad +\mu \zeta \left( \frac{Z}{Z^*}\right) \Lambda \rho _P=\mu \zeta \left( \frac{Z}{Z^*}\right) \Lambda \rho _P,\\&\quad \frac{\partial _\tau \rho _D+\mu \Lambda \rho _D}{\rho _D}=\mu \frac{\Lambda \rho _P}{\rho _P}=-\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) \end{aligned}$$

and (5.41) is proved.

Proof of (5.42). Recall (2.23)

$$\begin{aligned} \partial _\tau \rho _{\mathrm{Tot}}=-\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}-\frac{\mu \ell (r-1)}{2}\rho _{\mathrm{Tot}}-\left( 2\partial _Z\Psi _{\mathrm{Tot}}+\mu Z\right) \partial _Z\rho _{\mathrm{Tot}} \end{aligned}$$

which yields

$$\begin{aligned} \left| \frac{\partial _\tau \rho _{\mathrm{Tot}}+\mu \Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{\mu \ell (r-1)}{2}\right| =\left| -\Delta \Psi _{\mathrm{Tot}}-2\frac{\partial _Z\Psi _{\mathrm{Tot}}\partial _Z\rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right| \end{aligned}$$

and (5.42) follows from (5.35).

\(\Psi ^{(k)}\) terms. Integrating by parts:

$$\begin{aligned}&\left| b^2\int \partial ^k\Delta \rho _{D}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\right| \lesssim b^2\int \chi \rho ^2_T\frac{|\nabla \Psi ^{(k)}|}{\langle Z\rangle ^{k+3}}\nonumber \\&\quad \lesssim \int \chi \frac{\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2}{\langle Z\rangle }+b^4\int \chi \frac{\rho _{\mathrm{Tot}}^2}{\langle Z\rangle ^{2(k+3)-1}}\lesssim e^{-c_{k_m}\tau }, \end{aligned}$$
(5.44)

where we used (5.28).

Next

$$\begin{aligned} \mu (r-2)\int \Psi ^{(k)}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})=-\mu (r-2)\int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2.\nonumber \\ \end{aligned}$$
(5.45)

Similarly, using \(\partial _ZH_2=O\left( \frac{1}{\langle Z\rangle ^r}\right) \):

$$\begin{aligned}&k\int H_2\Psi ^{(k)}\nabla \cdot (\chi \rho _{\mathrm{Tot}}^2\nabla \Psi ^{(k)})\nonumber \\&\quad = -k\left[ \int \chi \mu \left[ 1+O\left( \frac{1}{\langle Z\rangle ^{\frac{1}{2}}}\right) \right] \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right. \nonumber \\&\qquad \left. +O\left( \int \chi \rho _{\mathrm{Tot}}^2\frac{|\Psi ^{(k)}|^2}{\langle Z\rangle ^{2r-\frac{1}{2}}}\right) \right] \nonumber \\&\quad = -k\mu \int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2+O\left( e^{-c_{k_m}\tau }\right) , \end{aligned}$$
(5.46)

where we also used that \(r>2\), \(k\ne 0\) and

$$\begin{aligned} \int \chi _{k,k,\sigma (k)}\rho _{\mathrm{Tot}}^2\frac{|\Psi ^{(k)}|^2}{\langle Z\rangle ^{2r-\frac{1}{2}}}\lesssim \int \chi _{k,k-1,\sigma (k)}\rho _{\mathrm{Tot}}^2\frac{|\nabla \Psi _{k-1}|^2}{\langle Z\rangle ^{2r-\frac{1}{2}-2}}\le \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma (k)+\frac{1}{2}} \end{aligned}$$

Then using (5.35):

$$\begin{aligned}&\left| \int 2\chi \rho _{\mathrm{Tot}}^2\nabla \Psi \cdot \nabla \Psi ^{(k)}\left( 2\frac{\nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{\nabla \chi }{\chi }\right) \cdot \nabla \Psi ^{(k)}\right| \nonumber \\&\quad \lesssim \int \chi \frac{\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2}{\langle Z\rangle } \lesssim e^{-c_{k_m}\tau } \end{aligned}$$
(5.47)

and from (5.27), (5.35):

$$\begin{aligned} \left| \int 2\chi \rho _{\mathrm{Tot}}\nabla \Psi \cdot \nabla \Psi ^{(k)}(\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)})\right|\lesssim & {} \int \chi |\nabla \Psi ^{(k)}|^2\left( |\partial (\rho _{\mathrm{Tot}}^2\nabla \Psi )|+\frac{|\rho _{\mathrm{Tot}}^2\nabla \Psi |}{\langle Z\rangle }\right) \nonumber \\\lesssim & {} \int \chi \frac{\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2}{\langle Z\rangle ^2}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.48)

We now carefully compute from (5.27) again:

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}} H_2\Lambda \Psi ^{(k)}\left( 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right) \nonumber \\&\qquad +\int H_2\Lambda \Psi ^{(k)}\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)}\nonumber \\&\quad = 2\sum _{i,j}\int \chi \rho _{\mathrm{Tot}} H_2Z_j\partial _j\Psi ^{(k)}\partial _i\rho _{\mathrm{Tot}}\partial _i\Psi ^{(k)}\nonumber \\&\quad -\sum _{i,j}\int \partial _i(\chi Z_j H_2\rho _{\mathrm{Tot}}^2)\partial _i\Psi ^{(k)}\partial _j\Psi ^{(k)}\nonumber \\&\qquad + \frac{1}{2}\int \nabla \cdot (\chi Z H_2\rho _{\mathrm{Tot}}^2)|\nabla \Psi ^{(k)}|^2+ \sum _{i,j} H_2\rho _{\mathrm{Tot}}^2Z_j\partial _j\Psi ^{(k)}\partial _i\chi \partial _i\Psi ^{(k)}\nonumber \\&\quad = \sum _{i,j}H_2\partial _j\Psi ^{(k)}\partial _i\Psi ^{(k)}\Bigg [2\chi \rho _{\mathrm{Tot}}\partial _i\rho _{\mathrm{Tot}}Z_j-\partial _i\chi Z_j\rho _{\mathrm{Tot}}^2\nonumber \\&\qquad -\chi \delta _{ij}\rho _{\mathrm{Tot}}^2-2\chi Z_j\rho _{\mathrm{Tot}}\partial _i\rho _{\mathrm{Tot}} -\chi Z_j\frac{\partial _iH_2}{H_2}\rho _{\mathrm{Tot}}^2+Z_j\rho _{\mathrm{Tot}}^2\partial _i\chi \Bigg ]\nonumber \\&\qquad + \frac{1}{2}\int \chi H_2\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\left[ d+\frac{\Lambda \chi }{\chi }+\frac{\Lambda H_2}{H_2}+2\frac{\Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right] \nonumber \\&\quad = \frac{1}{2}\mu \int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\left[ d-2+\frac{\Lambda \chi }{\chi }+2\frac{\Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+O\left( \frac{1}{\langle Z\rangle ^r}\right) \right] .\nonumber \\ \end{aligned}$$
(5.49)

Hence the final formula recalling (5.42):

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}} H_2\Lambda \Psi ^{(k)}\left( 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}+\rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\right) \nonumber \\&\qquad +\int H_2\Lambda \Psi ^{(k)}\rho _{\mathrm{Tot}}^2\nabla \chi \cdot \nabla \Psi ^{(k)} +\int \chi \partial _\tau \rho _{\mathrm{Tot}}\rho _{\mathrm{Tot}}|\nabla \Psi ^{(k)}|^2\nonumber \\&\quad = \int \mu \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\left[ \frac{d-2}{2}+\frac{1}{2}\frac{\Lambda \chi }{\chi }+\frac{\Lambda \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{1}{\mu }\frac{\partial _\tau \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+O\left( \frac{1}{\langle Z\rangle ^r}\right) \right] \nonumber \\&\quad = \int \mu \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\left[ \frac{d-2}{2}+\frac{1}{2}\frac{\Lambda \chi }{\chi }-\frac{2(r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) \right] \nonumber \\&\quad = \int \mu \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\left[ \frac{d-2}{2}+\frac{1}{2}\frac{\Lambda \chi }{\chi }-\frac{2(r-1)}{p-1}\right] +O(e^{-c_{k_m}\tau }). \end{aligned}$$
(5.50)

Loss of derivatives terms. We integrate by parts the non linear term which must loose derivatives:

$$\begin{aligned}&b^2\left| \int \rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)}\Delta {\tilde{\rho }}^{(k)}\right| \nonumber \\&\quad \lesssim b^2\left| \int \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\right| +b^2\int \chi \frac{\rho _{\mathrm{Tot}}}{\langle Z\rangle ^2}|\nabla \Psi ^{(k)}||\nabla {\tilde{\rho }}^{(k)}|\nonumber \\&\quad \lesssim b^3\int \chi |\nabla {\tilde{\rho }}^{(k)}|^2+b\left[ \int \chi \frac{\rho _{\mathrm{Tot}}^2|\Delta \Psi ^{(k)}|^2}{\langle Z\rangle ^2}+\int \chi \frac{\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2}{\langle Z\rangle ^4}\right] \nonumber \\&\quad \lesssim e^{-c_{k_m}\tau } + b\int \chi \frac{\rho _{\mathrm{Tot}}^2|\Delta \Psi ^{(k)}|^2}{\langle Z\rangle ^2}. \end{aligned}$$
(5.51)

We now use (5.20) for \(0\le k\le k_m-1\) which implies

$$\begin{aligned} \int \chi _{k,k,\sigma (k)}\frac{\rho _{\mathrm{Tot}}^2|\Delta \Psi ^{(k)}|^2}{\langle Z\rangle ^2}\le & {} \int \chi _{k+1,k+1,\sigma (k+1)}\rho _{\mathrm{Tot}}^2|\Delta \Psi ^{(k)}|^2\\\lesssim & {} \Vert {\tilde{\rho }},\Psi \Vert ^2_{k+1,\sigma (k+1)}\lesssim 1. \end{aligned}$$

Hence

$$\begin{aligned}&b^2\left| \int \rho _{\mathrm{Tot}}\nabla \chi \cdot \nabla \Psi ^{(k)} \Delta {\tilde{\rho }}^{(k)}\right| +b^2\left| \int \rho _{\mathrm{Tot}}\Delta \Psi ^{(k)}\nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\right| \nonumber \\&\quad +b^2\left| \int \nabla \chi \cdot \nabla {\tilde{\rho }}^{(k)}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}\right| \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$
(5.52)

Conclusion for linear terms. The energy identity (5.10) with the weight \(\chi \) in (5.22), together with the estimates (5.29)–(5.34), (5.36)–(5.39), (5.43)–(5.48), (5.50) and (5.52) yields:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int b^2\chi |\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} \nonumber \\&\quad \le e^{-c_{k_m\tau }} + \mu \int \chi \left[ -k+\frac{d}{2}-(r-1)-\frac{2(r-1)}{p-1}+\frac{1}{2}\frac{\mu ^{-1}\partial _\tau \chi +\Lambda \chi }{\chi }\right] \nonumber \\&\qquad \times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \nonumber \\&\qquad + \int F_1\chi (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^{(k)}+b^2\int \chi \nabla F_1\cdot \nabla {\tilde{\rho }}^{(k)}\nonumber \\&\qquad +\int \chi \rho ^2_T\nabla F_2\cdot \nabla \Psi ^{(k)}. \end{aligned}$$
(5.53)

Step 4 \(F_1\) terms. We recall (5.7) and claim the bound:

$$\begin{aligned}&(p-1)\int \chi F_1^2\rho _D^{p-2}\rho _{\mathrm{Tot}}+b^2\int \chi |\nabla F_1|^2\nonumber \\&\quad \lesssim e^{-c_{k_m}\tau }\left[ 1+\Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma }\right] \end{aligned}$$
(5.54)

Source term induced by localization. Recall (5.2)

$$\begin{aligned} \tilde{{\mathcal {E}}}_{P,\rho }= & {} \partial _\tau \rho _D+\rho _D\left[ \Delta \Psi _P+\mu \frac{\ell (r-1)}{2}+\left( 2\partial _Z\Psi _P+ \mu Z\right) \frac{\partial _Z\rho _D}{\rho _D}\right] \\= & {} \partial _\tau \rho _D+\mu \Lambda \rho _D+\mu \frac{\ell (r-1)}{2}\rho _D+\rho _D\Delta \Psi _P+2\partial _Z\Psi _P\partial _Z\rho _D. \end{aligned}$$

From the proof of (5.41)

$$\begin{aligned} \rho _D=\zeta \left( \frac{Z}{Z^*}\right) \rho _P,\quad \partial _\tau \rho _D+\mu \Lambda \rho _D=\mu \zeta \left( \frac{Z}{Z^*}\right) \Lambda \rho _P. \end{aligned}$$

Therefore, using the profile equation for \(\rho _P\), we obtain

$$\begin{aligned} \tilde{{\mathcal {E}}}_{P,\rho }= 2\frac{\Psi '_P}{Z} \frac{Z}{Z^*}\zeta '\rho _P. \end{aligned}$$

From (2.10) and (2.19) we then conclude that

$$\begin{aligned} |\partial ^k\tilde{{\mathcal {E}}}_{P,\rho }|\lesssim \frac{\rho _D}{\langle Z\rangle ^{k+r}}{} \mathbf{1}_{Z\ge Z^*}. \end{aligned}$$
(5.55)

Hence, recalling (5.19) and ():

$$\begin{aligned}&\int \chi \rho _{D}^{p-2}\rho _{\mathrm{Tot}}|\partial ^k\tilde{{\mathcal {E}}}_{P,\rho }|^2\\&\quad \lesssim \int _{Z\ge Z^*}\frac{Z^{d-1}dZ}{Z^{-2k+2\left( \frac{d}{2}-(r-1)+{\tilde{\nu }}-\frac{2(r-1)}{p-1}\right) }}\frac{1}{Z^{2k+2r}}\frac{(Z)^{-\frac{2(r-1)(p+1)}{p-1}}}{\left( \frac{Z}{Z^*}\right) ^{(p-1)n_P-2(r-1)+2(r-2)-4{\tilde{\nu }}}}\\&\quad \lesssim \int _{Z\ge Z^*}\frac{dZ}{Z^{2r+2{\tilde{\nu }}+1}\left( \frac{Z}{Z^*}\right) ^{(p-1)n_P-2-4{\tilde{\nu }}}}\lesssim (Z^*)^{-2r-2{\tilde{\nu }}}\lesssim e^{-c\tau }. \end{aligned}$$

Similarly, from ():

$$\begin{aligned} b^2\int \chi |\nabla \partial ^k \tilde{{\mathcal {E}}}_{P,\rho }|^2\lesssim \int _{Z\ge Z^*}\frac{Z^{d-1}dZ}{Z^{2\left( \frac{d}{2}-1-{\tilde{\nu }}\right) +2+2r}} \lesssim (Z^*)^{-2r+2{\tilde{\nu }}}\lesssim e^{-c\tau }. \end{aligned}$$

\([\partial ^k,H_1]\) term. We use (5.5) to estimate:

$$\begin{aligned} \left| \begin{array}{ll} |[\partial ^k,H_1]{\tilde{\rho }}|\lesssim \sum _{j=0}^{k-1}|\partial ^j{\tilde{\rho }}\partial ^{k-j}H_1|\lesssim \sum _{j=0}^{k-1}\frac{|\partial ^j{\tilde{\rho }}|}{\langle Z\rangle ^{r+k-j}},\\ |\nabla [\partial ^k,H_1]{\tilde{\rho }}|\lesssim \sum _{j=0}^{k}\frac{|\partial ^j{\tilde{\rho }}|}{\langle Z\rangle ^{1+r+k-j}}. \end{array}\right. \end{aligned}$$
(5.56)

Hence

$$\begin{aligned} (p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}([\partial ^k,H_1]{\tilde{\rho }})^2\lesssim & {} \sum _{j=0}^{k-1}\int \chi \rho _D^{p-1} \frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\\\lesssim & {} \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma (k)+r}\lesssim e^{-c_{k_m}\tau }, \end{aligned}$$

and

$$\begin{aligned}&b^2\int \chi |\nabla ([\partial ^k,H_1]{\tilde{\rho }})|^2\lesssim b^2 \sum _{j=0}^{k}\int \chi \frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(1+r+k-j)}}\\&\quad \lesssim b^2\int \chi \frac{{\tilde{\rho }}^2}{\langle Z\rangle ^{2(1+r+k)}}+b^2 \sum _{j=0}^{k}\int \chi \frac{|\partial ^j\nabla {\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\\&\quad \lesssim b^2\int \chi \frac{\rho _D^2}{\langle Z\rangle ^{2(1+r+k)}}+ e^{-c_{k_m}\tau }\lesssim e^{-c_{k_m}\tau }, \end{aligned}$$

where we used the bootstrap bound (4.40), the decay of \(b^2\) and ().

\([\partial ^k,H_2]\) term. Similarly, from (5.5):

$$\begin{aligned} \left| \begin{array}{ll} |[\partial ^k,H_2]\Lambda {\tilde{\rho }}|\lesssim \sum _{j=0}^{k-1}| \partial ^j(\Lambda {\tilde{\rho }})\partial ^{k-j}H_2|\lesssim \sum _{j=1}^{k}\frac{|\partial ^j{\tilde{\rho }}|}{\langle Z\rangle ^{r-1+k-j}}\\ |\nabla [\partial ^k,H_2]\Lambda {\tilde{\rho }}|\lesssim \sum _{j=1}^{k+1}\frac{|\partial ^j{\tilde{\rho }}|}{\langle Z\rangle ^{r+k-j}}. \end{array}\right. \end{aligned}$$
(5.57)

Hence, using \(r>1\):

$$\begin{aligned}&(p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}([\partial ^k,H_2]\Lambda {\tilde{\rho }})^2\\&\quad \lesssim \sum _{j=1}^{k}\int {\chi }\rho _D^{p-1} \frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r-1+k-j)}} \lesssim e^{-c_{k_m}\tau }, \end{aligned}$$

and

$$\begin{aligned}&b^2\int \chi |\nabla ([\partial ^k,H_2]\Lambda {\tilde{\rho }})|^2\lesssim b^2 \sum _{j=1}^{k+1}\int \chi \frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\\&\quad = b^2 \sum _{j=0}^{k}\int \chi \frac{|\partial ^j\nabla {\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r-1+k-j)}}\lesssim \Vert {\tilde{\rho }},\Psi \Vert ^2_{k,\sigma +r-1}\lesssim e^{-c_{k_m}\tau }\ \end{aligned}$$

Nonlinear term. Changing indices, we need to estimate terms

$$\begin{aligned} N_{j_1,j_2}=\partial ^{j_1}\rho _{\mathrm{Tot}}\partial ^{j_2}\nabla \Psi , \ \ j_1+j_2=k+1, \ \ 2\le j_1,j_2\le k-1. \end{aligned}$$
(5.58)

For the profile term:

$$\begin{aligned} |\partial ^{j_1}\rho _D\partial ^{j_2}\nabla \Psi |\lesssim \rho _D\frac{|\partial ^{j_2}\nabla \Psi |}{\langle Z\rangle ^{j_1}}= \rho _D\frac{|\partial ^{j_2}\nabla \Psi |}{\langle Z\rangle ^{k+1-j_2}} \end{aligned}$$

and hence using from (5.19) the rough global bound:

$$\begin{aligned} \rho _D\lesssim \frac{1}{\langle Z\rangle ^{\frac{2(r-1)}{p-1}}} \end{aligned}$$
(5.59)

yields

$$\begin{aligned} \int (p-1) \chi N_{j_1,j_2}^2\rho _D^{p-2}\rho _{\mathrm{Tot}}\lesssim & {} \int \chi \frac{\rho _{\mathrm{Tot}}^2|\partial ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)+2(r-1)}} \\= & {} \int \chi \frac{\rho _{\mathrm{Tot}}^2|\partial ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k-j_2)+2r}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

Similarly, after taking a derivative:

$$\begin{aligned} b^2\int \chi |\nabla N_{j_1,j_2}|^2\lesssim b^2\int \chi \frac{\rho _{\mathrm{Tot}}^2|\partial ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+2-j_2)}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

We now turn to the control of the nonlinear term. If \(j_1\le \frac{4k_m}{9}\), then from (4.40):

$$\begin{aligned} \int \chi \rho _D^{p-1}|\partial ^{j_1}{\tilde{\rho }}\partial ^{j_2}\nabla \Psi |^2\lesssim \int \chi \rho _D^2\frac{|\partial ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)+2(r-1)}}\lesssim e^{-c_{k_m}\tau }.\qquad \end{aligned}$$
(5.60)

If \(j_2\le \frac{4k_m}{9}\), then from (4.40) and \(b=\frac{1}{(Z^*)^{r-2}}\):

$$\begin{aligned} \int \chi \rho _D^{p-1}|\partial ^{j_1}{\tilde{\rho }}\partial ^{j_2}\nabla \Psi |^2\lesssim & {} \int _{Z\le Z^*} \chi \rho _D^{p-1}\frac{|\partial ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k+1+(r-2)-j_1)}}\\&+b^2\int _{Z\ge Z^*}\chi \rho _D^{p-1}\frac{|\partial ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k+1-j_1)}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

We may therefore assume \(j_1,j_2\ge m_0=\frac{4k_m}{9}+1\), which implies \(k\ge m_0\) and \(j_1,j_2\le \frac{2k_m}{3}\). From (4.29):

$$\begin{aligned} \sigma (k)= & {} -\alpha (k_m-k)\ge -\alpha \left( k_m-\frac{4k_m}{9}\right) =-\frac{4}{5}\left( 1-\frac{4}{9}\right) k_m+O_{k_m\rightarrow +\infty }(1)\nonumber \\\ge & {} -\frac{4k_m}{9}+O_{k_m\rightarrow +\infty }(1). \end{aligned}$$
(5.61)

From (4.41):

$$\begin{aligned} \sigma (k)+n(j_1)+n(j_2)\ge -\frac{4k_m}{9}+\frac{k_m}{4}+\frac{k_m}{4}+O_{k_m\rightarrow +\infty }(1)\ge \frac{k_m}{20}\nonumber \\ \end{aligned}$$
(5.62)

and hence from (4.40) and interpolating on \(Z\le Z^*_c\) with (4.39):

$$\begin{aligned} \int \chi \rho _D^{p-1}|\partial ^{j_1}{\tilde{\rho }}\partial ^{j_2}\nabla \Psi |^2\lesssim & {} e^{-c_{k_m}\tau } +\int _{Z\ge Z^*_c}\frac{Z^{d-1}dZ}{\langle Z\rangle ^{\frac{k_m}{10}}}\\&\quad \left[ \mathbf{1}_{Z\le Z^*}+\left( \frac{Z}{Z^*}\right) ^{-(p-3)n_P-\frac{4(r-1)}{(p-1)} -2(r-2)-2(r-1)+4{\tilde{\nu }}} \mathbf{1}_{Z\ge Z^*}\right] \\\lesssim & {} e^{-c_{k_m}\tau } \end{aligned}$$

The \(b^2\) derivative term and the other nonlinear term in (5.7) are estimated similarly. We note that the relation

$$\begin{aligned} k_m\gg n_P\gg 1 \end{aligned}$$

ensures that the terms containing \(k_m\) are dominant and eliminates the need to track the dependence on \(n_P\). This concludes the proof of (5.54).

Step 5 \(F_2\) terms. We claim:

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|\nabla F_2|^2 \lesssim e^{-c_{k_m}\tau }\left[ 1+\Vert {\tilde{\rho }},\Psi \Vert ^2_{k+1,\sigma (k+1)}\right] . \end{aligned}$$
(5.63)

Source term induced by localization. Recall (5.2):

$$\begin{aligned} \tilde{{\mathcal {E}}}_{P,\Psi } =|\nabla \Psi _P|^2+\rho _D^{p-1}+e\Psi _P+\frac{1-{\mathscr {e}}}{2}\Lambda \Psi _P-1=\rho _D^{p-1}-\rho _P^{p-1} \end{aligned}$$

which yields the rough bound

$$\begin{aligned} |\nabla \partial ^k\tilde{{\mathcal {E}}}_{P,\Psi }|\lesssim \frac{1}{\langle Z\rangle ^{k+1+2(r-1)}}{} \mathbf{1}_{Z\ge Z^*} \end{aligned}$$

and hence, from (),

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|\nabla \partial ^k\tilde{{\mathcal {E}}}_{P,\Psi }|^2\lesssim & {} \int _{Z\ge Z^*}\frac{Z^{d-1}}{b^2 \langle Z\rangle ^{-2k+2\left( \frac{d}{2}+{\tilde{\nu }}-1\right) }}\frac{\left( \frac{Z}{Z^*}\right) ^{4{\tilde{\nu }}}}{\langle Z\rangle ^{2k+2+4(r-1)}}\\\lesssim & {} {Z^*}^{2(r-2)-4(r-1)+4{\tilde{\nu }}}\lesssim e^{-c\tau } \end{aligned}$$

\([\partial ^k,H_2]\Lambda \Psi \) term. From (5.5):

$$\begin{aligned} |\nabla ([\partial ^k,H_2]\Lambda \Psi )|\lesssim \sum _{j=1}^{k+1}\frac{|\partial ^j\Psi |}{\langle Z\rangle ^{r+k-j}}\lesssim \sum _{j=0}^{k}\frac{|\nabla \partial ^j\Psi |}{\langle Z\rangle ^{r+k-j-1}} \end{aligned}$$

and hence

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|\nabla ([\partial ^k,H_2]\Lambda \Psi )|^2\lesssim \sum _{j=0}^{k}\int \chi \rho _{\mathrm{Tot}}^2\frac{|\nabla \partial ^j\Psi |^2}{\langle Z\rangle ^{2(r-1+k-j)}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

\([\partial ^k,\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\) term. By Leibnitz:

$$\begin{aligned} \left| \left[ [\partial ^k,\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right] \right| \lesssim \sum _{j=0}^{k-2}\frac{|\partial ^j{\tilde{\rho }}|}{\langle Z\rangle ^{k-j}}\rho _D^{p-2} \end{aligned}$$

and, hence, taking a derivative:

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}}^2\left| \nabla \left[ [\partial ^k,\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\partial \rho _D\partial ^{k-1}{\tilde{\rho }}\right] \right| ^2\\&\quad \lesssim \sum _{j=0}^{k-1}\int \chi \rho _D^{2(p-2)+2}\frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-j)+2}}\\&\quad \lesssim \sum _{j=0}^{k-1}\int \chi \rho _D^{p-1}\frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-j)+2}} \lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

Nonlinear \(\Psi \) term. Let

$$\begin{aligned} \partial N_{j_1,j_2}=\partial ^{j_1}\nabla \Psi \partial ^{j_2}\nabla \Psi , \ \ j_1+j_2=k+1, \ \ j_1,j_2\ge 1. \end{aligned}$$

If \(j_1\le \frac{4k_m}{9},\) then from (4.40):

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|\nabla N_{j_1,j_2}|^2\lesssim \int \rho _{\mathrm{Tot}}^2\chi \frac{|\partial ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)}}\lesssim \Vert {\tilde{\rho }},\Psi \Vert _{k,\sigma +\frac{1}{2}}^2\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

The expression being symmetric in \(j_1,j_2\), we may assume \(j_1,j_2\ge m_0=\frac{4k_m}{9}+1\), \(j_1,j_2\le \frac{2k_m}{3}\) and \(k\ge m_0=\frac{4k_m}{9}+1\). Using (4.40), (5.62) and arguing as above (\(k_m\gg n_P\)):

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|\nabla N_{j_1,j_2}|^2\lesssim & {} e^{-c\tau }+\int _{Z\ge Z^*_c}\frac{dZ}{\langle Z\rangle ^{\frac{k_m}{10}}}\left[ \mathbf{1}_{Z\le Z^*}+\left( \frac{Z}{Z^*}\right) ^{2n_P+4{\tilde{\nu }}}\mathbf{1}_{Z\ge Z^*}\right] \\\lesssim & {} e^{-c_{k_m}\tau }. \end{aligned}$$

Quantum pressure term. We estimate from Leibniz:

$$\begin{aligned}&b^2\left| \partial ^k\left( \frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right) -\frac{\partial ^k\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{k\partial ^{k-1}\Delta \rho _{\mathrm{Tot}}\partial \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right| \\&\quad \lesssim b^2\sum _{j_1+j_2=k,j_2\ge 2}\left| \partial ^{j_1}\Delta \rho _{\mathrm{Tot}}\partial ^{j_2}\left( \frac{1}{\rho _{\mathrm{Tot}}}\right) \right| . \end{aligned}$$

and using the Faa-di Bruno formula:

$$\begin{aligned} \left| \partial ^{j_2}\left( \frac{1}{\rho _{\mathrm{Tot}}}\right) \right| \lesssim \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{q_1+2q_2+\cdots +j_2q_{j_2}=j_2}\Pi _{i=1}^{j_2}|(\partial ^i\rho _{\mathrm{Tot}})^{q_i}|. \end{aligned}$$

We decompose \(\rho _{\mathrm{Tot}}=\rho _D+{\tilde{\rho }}\) and control the \(\rho _D\) term using the bound

$$\begin{aligned} |\partial ^i\rho _D|\lesssim \frac{\rho _D}{\langle Z\rangle ^i} \end{aligned}$$

which yields

$$\begin{aligned} \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{m_1+2m_2+\cdots +j_2m_{j_2}=j_2}\Pi _{i=1}^{j_2}|(\partial ^i\rho _D)^{m_i}| \lesssim \frac{1}{\rho _{\mathrm{Tot}}\langle Z\rangle ^{j_2}} \end{aligned}$$
(5.64)

and hence the corresponding contribution to (5.63):

$$\begin{aligned}&b^4\int \chi \rho _{\mathrm{Tot}}^2\left\{ \sum _{j_1+j_2=k,j_2\ge 2}\frac{|\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2}{\rho _{\mathrm{Tot}}^2\langle Z\rangle ^{2j_2}}+\frac{|\partial ^{j_1}\Delta \rho _{\mathrm{Tot}}|^2}{\rho _{\mathrm{Tot}}^2\langle Z\rangle ^{2j_2+2}}\right\} \\&\quad \lesssim b^4\sum _{j_1+j_2=k,j_2\ge 2}\left[ \int \chi \frac{\rho _D^2dZ}{\langle Z\rangle ^{2j_2+2(j_1+3)}}+\int \chi \frac{|\partial ^{j_1+3}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2j_2}}\right] \\&\quad \lesssim b^4\int \chi \frac{\rho _D^2dZ}{\langle Z\rangle ^{2k+6}}+ b^4\sum _{j_1=2}^k\int \chi \frac{|\nabla \partial ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-j_1)+2}}\lesssim e^{-c_{k_m}\tau } \end{aligned}$$

where we used (5.28) in the last step.

We now turn to the control of the nonlinear term and consider

$$\begin{aligned} N_{j_1,j_2}=b^2\left( \partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}\right) \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{q_1+2q_2+\cdots +j_2q_{j_2}=j_2}\Pi _{i=1}^{j_2}(\partial ^i{\hat{\rho }})^{q_i}, \end{aligned}$$

where \({\hat{\rho }}\) is either \(\rho _D\) or \({\tilde{\rho }}\). In both cases we will use the weaker estimates (4.40).

First assume that \(q_i=0\) whenever \(i\ge \frac{4k_m}{9}+1\), then from (4.40):

$$\begin{aligned} |N_{j_1,j_2}|\lesssim & {} b^2|\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|\frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{q_1+2q_2+\cdots +j_2q_{j_2}=j_2}\Pi _{i=1}^{j_2}|(\partial ^i{\hat{\rho }})^{q_i}|\\\lesssim & {} b^2 \frac{|\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|}{\rho _{\mathrm{Tot}}\langle Z\rangle ^{j_2}} \end{aligned}$$

and the conclusion follows verbatim as above. Otherwise, there are at most two value \(\frac{4k_m}{9}\le i_1\le i_2\le j_2\) with \(q_{i_1},q_{i_2}\ne 0\) and \(q_{i_1}+q_{i_2}\le 2\). Hence from (4.40):

$$\begin{aligned} \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\Pi _{i=1}^{j_2}|(\partial ^i{\hat{\rho }})^{q_i}|\lesssim & {} \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}|\partial ^{i_1}{\hat{\rho }}|^{q_{i_1}}|\partial ^{i_2}{\hat{\rho }}|^{q_{i_2}}\Pi _{1\le i\le j_2, i\notin \{i_1,i_2\}}\left( \frac{\rho _D}{\langle Z\rangle ^{i}}\right) ^{q_i}\\\lesssim & {} \left( \frac{|\partial ^{i_1}{\hat{\rho }}|}{\rho _D}\right) ^{q_{i_1}}\left( \frac{|\partial ^{i_2}{\hat{\rho }}|}{\rho _D}\right) ^{q_{i_2}}\frac{1}{\rho _{\mathrm{Tot}}\langle Z\rangle ^{j_2-(q_{i_1}i_1+q_{i_2}i_2)}}. \end{aligned}$$

Assume first \(i_2\ge \frac{2k_m}{3}+1\), then \(q_{i_1}=0\), \(q_{i_2}=1\) and \(j_1+3\le \frac{4k_m}{9}\) from which:

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \chi \rho _{\mathrm{Tot}}^2 |\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\rho ^2_T}\frac{1}{\rho ^2_T\langle Z\rangle ^{2(j_2-i_2)}}\\\lesssim & {} b^4\int \chi \frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(j_2-i_2)+2(j_1+3)}}\\\lesssim & {} b^4\int \chi \frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(k-i_2)+6}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

There remains the case \(\frac{4k_m}{9}+1\le i_1\le i_2\le \frac{2k_m}{3}\) which imply \(j_1+3\le \frac{2k_m}{3}\), and we distinguish cases:

case \((m_{i_1},m_{i_2})=(0,1)\): if \(j_1+3\le \frac{4k_m}{9}\), we estimate

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim b^4\int \chi \rho _D^2 |\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\rho ^2_T}\frac{1}{\rho ^2_T\langle Z\rangle ^{2(j_2-i_2)}}\\&\quad \lesssim b^4\int \chi \frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(j_2-i_2)+2(j_1+3)}} \lesssim b^4\int \chi \frac{|\partial ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(k-i_2)+6}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

Otherwise, \(\frac{4k_m}{9}+1\le j_1+3\le \frac{2k_m}{3}\). Since \(j_2\ge \frac{4k_m}{9}+1\), then necessarily \(j_2\le \frac{2k_m}{3}\). Hence \(\frac{4k_m}{9}+1\le j_1+3\le \frac{2k_m}{3}\), \(\frac{4k_m}{9}+1\le j_2\le \frac{2k_m}{3}\) and we estimate from (4.40):

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \frac{Z^{d-1}dZ}{\langle Z\rangle ^{2\left( \sigma (k)+\frac{k_m}{4}+\frac{k_m}{4}+j_2-i_2\right) }}\\&\left[ \mathbf{1}_{Z\le Z^*}+\left( \frac{Z}{Z^*}\right) ^{2n_P+4{\tilde{\nu }}}{} \mathbf{1}_{Z\ge Z^*}\right] \lesssim b^4\lesssim e^{-c_{k_m}\tau }, \end{aligned}$$

where we once again used that in this range of k

$$\begin{aligned} \sigma (k)+\frac{k_m}{2}\ge \frac{k_m}{20},\qquad k_m\gg n_P\gg 1 \end{aligned}$$

case \(m_{i_1}+m_{i_2}=2\): we use (4.40) and estimate crudely:

$$\begin{aligned} \int \chi \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \chi |\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\left( \frac{1}{\langle Z\rangle ^{\frac{k_m}{4}}}\right) ^4\\&\lesssim b^4\int \frac{Z^{d-1}dZ}{\langle Z\rangle ^{2\left( \sigma (k)+\frac{k_m}{2}\right) }}\\&\quad \times \left[ \mathbf{1}_{Z\le Z^*}+\left( \frac{Z}{Z^*}\right) ^{2n_P+4{\tilde{\nu }}}\mathbf{1}_{Z\ge Z^*}\right] \lesssim b^4\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

\(\text {NL}({\tilde{\rho }})\) term. We expand, using, according to our assumptions, that the power of the nonlinear term is an integer \(p\ge 3\):

$$\begin{aligned} \text {NL}({\tilde{\rho }})=(\rho _D+{\tilde{\rho }})^{p-1}-\rho _D^{p-1}-(p-1)\rho _D^{p-2}{\tilde{\rho }}=\sum _{q=2}^{p-1}c_{q}{\tilde{\rho }}^{q}\rho _D^{p-1-q} \end{aligned}$$

and hence by Leibniz:

$$\begin{aligned} \partial ^{k}\text {NL}({\tilde{\rho }})= & {} \sum _{q=2}^{p-1}\sum _{j_1+j_2=k}c_{q,j_1,j_2}\partial ^{j_1}({\tilde{\rho }}^{q})\partial ^{j_2}(\rho _D^{p-1-q})\\= & {} \sum _{q=2}^{p-1}\sum _{j_1+j_2=k}\sum _{\ell _1+\cdots +\ell _q=j_1}\partial ^{\ell _1}{\tilde{\rho }}\cdots \partial ^{\ell _q}{\tilde{\rho }}\partial ^{j_2}(\rho _D^{p-1-q}). \end{aligned}$$

Let

$$\begin{aligned} N_{\ell _1,\ldots ,\ell _q,j_1,q}=\partial ^{\ell _1}{\tilde{\rho }}\cdots \partial ^{\ell _q}{\tilde{\rho }}\partial ^{j_2}(\rho _D^{p-1-q}),\ \ \ell _1\le \cdots \le \ell _q, \end{aligned}$$

then

$$\begin{aligned} |\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim | N_{\ell _1,\ldots ,\ell _q,j_1,q}^{(1)}|+|N_{\ell _1,\ldots ,\ell _q,j_1,q}^{(2)}| \end{aligned}$$

with

$$\begin{aligned} | N^{(1)}_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim |\partial ^{m_1}{\tilde{\rho }}\cdots \partial ^{m_q}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}, \ \ \left| \begin{array}{ll} 0\le m_1\le \cdots \le m_q\le k+1,\\ m_1+\cdots +m_q=j_1+1.\end{array}\right. \end{aligned}$$

We estimate \(N^{(1)}_{\ell _1,\ldots ,\ell _q,j_1,q}\), the other term being estimated similarly. We distinguish cases.

case \(m_q\le \frac{4k_m}{9}\), then from (4.40):

$$\begin{aligned} | N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|\lesssim \frac{{\tilde{\rho }}^{q}}{\langle Z\rangle ^{j_1+1}}\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\lesssim \frac{\rho _D^{p-1}}{\langle Z\rangle ^{k+1}} \end{aligned}$$

and hence, from (5.19) and (5.25), the contribution of this term is given by

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}}^2|N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|^2\lesssim e^{-c\tau }\\&\qquad +\int _{Z\ge Z^*_c}\frac{Z^{d-1}dZ}{\langle Z\rangle ^{2\sigma (k)+2(k+1)}} \frac{\mathbf{1}_{Z\le Z^*}+\left( \frac{Z}{Z^*}\right) ^{ -2(p-1)n_P-{4(r-1)}-2(r-2)+4{\tilde{\nu }}}{} \mathbf{1}_{Z\ge Z^*}}{\langle Z\rangle ^{4(r-1)+\frac{4(r-1)}{p-1}}}\\&\quad \lesssim e^{-c\tau }+\int _{Z\ge Z^*_c}\frac{dZ}{\langle Z\rangle ^{2(r-1)+3}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

We now assume \(m_q\ge \frac{4k_m}{9}+1\) and recall \(m_q\le j_1+1\le k+1\le k_m\).

case \(m_{q-1}\le \frac{4k_m}{9}\), then from (4.40):

$$\begin{aligned} |N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|\lesssim \frac{\rho ^{q-1}_D}{\langle Z\rangle ^{j_1-m_q+1}}|\partial ^{m_q}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\lesssim |\partial ^{m_q}{\tilde{\rho }}|\rho _D^{p-2}\frac{1}{\langle Z\rangle ^{k+1-m_q}}. \end{aligned}$$

If \(m_q\le k\) then

$$\begin{aligned}&\int \chi \rho _{\mathrm{Tot}}^2|N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|^2\lesssim \int \chi \rho _D^2\frac{|\partial ^{m_q}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-m_q+1)+\frac{4(r-1)(p-2)}{p-1}}} \lesssim e^{-c_{k_m}\tau } \end{aligned}$$

On the other hand, if \(m_q=k+1\), then, using (5.20)

$$\begin{aligned}&\int \chi _{k,k,\sigma (k)} \rho _{\mathrm{Tot}}^2|N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|^2\lesssim \int \chi _{k,k,\sigma (k)} \rho _D^2\rho _D^{2(p-3)}{{\tilde{\rho }}^2\,|\partial ^{k+1}{\tilde{\rho }}|^2}\\&\quad \lesssim \int _{Z<Z_c^*} \chi _{k+1,k+1,\sigma (k+1)} \rho _D^{2(p-2)}\langle Z\rangle ^2{{\tilde{\rho }}^2\,|\partial ^{k+1}{\tilde{\rho }}|^2}\\&\qquad + \int _{Z>Z_c^*}\chi _{k+1,k+1,\sigma (k+1)} \rho _D^{p-1}\frac{\,|\partial ^{k+1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{-2+2(r-1)}}\\&\quad \lesssim e^{-c_{k_m}\tau } \Vert {\tilde{\rho }},\Psi \Vert ^2_{k+1,\sigma (k+1)}, \end{aligned}$$

where we used the following interpolation bound

$$\begin{aligned} \Vert {\tilde{\rho }}\Vert _{L^\infty (Z\le Z^*_c)}\lesssim e^{-c\tau }, \end{aligned}$$

the estimate

$$\begin{aligned} {\tilde{\rho }}\lesssim \rho _D\lesssim \langle Z\rangle ^{-\frac{2(r-1)}{p-1}} \end{aligned}$$

and the condition

$$\begin{aligned} -2+2(r-1)>0, \end{aligned}$$

which follows from \(r>2\).

case \(m_{q-1}\ge \frac{4k_m}{9}+1\), then necessarily \(m_{q-2}\le \frac{4k_m}{9}<\frac{4k_m}{9}+1\le m_{q-1}\le m_{q}<\frac{2k_m}{3}\) and \(k\ge \frac{4k_m}{9}+1\). Hence

$$\begin{aligned} |N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|\lesssim \frac{\rho ^{q}_D}{\langle Z\rangle ^{\frac{k_m}{4}+\frac{k_m}{4}}}\rho _D^{p-1-q}\lesssim \frac{\rho _D^{p-1}}{\langle Z\rangle ^{\frac{k_m}{2}}}. \end{aligned}$$

The integral for \(Z<Z^*_c\) is estimated as above, and we further estimate from (5.25) and (5.62), using that \(k_m\gg n_P\gg 1\),

$$\begin{aligned} \int _{Z\ge Z^*_c}\chi \rho _D^2 |N^{(1)}_{m_1,\ldots ,m_q,j_1,q}|^2\lesssim \int _{Z\ge Z_c^*}\frac{dZ}{\langle Z\rangle ^{\frac{k_m}{20}}}\lesssim e^{-c_{k_m}\tau }. \end{aligned}$$

This concludes the proof of (5.63).

Step 6 Conclusion. Injecting (5.54) and (5.63) into (5.53) yields:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int b^2\chi |\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\int \chi \rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\int \chi \rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right\} \\&\quad \le \mu \int \chi \left[ -k+\frac{d}{2}-(r-1)-\frac{2(r-1)}{p-1}+\frac{1}{2}\frac{\mu ^{-1}\partial _\tau \chi +\Lambda \chi }{\chi }\right] \\&\qquad \times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] + e^{-c_{k_m\tau }}. \end{aligned}$$

We now compute, noting that \(\partial _\tau Z^*=\mu Z^*\) and that \(\xi _k\) only depends on \(\tau \) through \(Z^*\):

$$\begin{aligned}&\partial _\tau \chi +\mu \Lambda \chi \\&\quad = \frac{1}{\langle Z\rangle ^{2\sigma (k)}}\left[ \partial _\tau \xi _k\left( \frac{Z}{Z^*}\right) +\mu \Lambda \xi _k\left( \frac{Z}{Z^*}\right) \right] + \xi _k\left( \frac{Z}{Z^*}\right) \mu \Lambda \left( \frac{1}{\langle Z\rangle ^{2\sigma (k)}}\right) \\&\quad = \xi _k\left( \frac{Z}{Z^*}\right) \mu \Lambda \left( \frac{1}{\langle Z\rangle ^{2\sigma (k)}}\right) . \end{aligned}$$

Hence recalling (5.23):

$$\begin{aligned}&k-\frac{d}{2}+r-1+\frac{2(r-1)}{p-1}-{\frac{1}{2}\frac{\mu ^{-1}\partial _\tau \chi +\Lambda \chi }{\chi }}\\&\quad =k+\sigma (k)-\frac{d}{2}+r-1+\frac{2(r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle }\right) \\&\quad \ge \sigma _\nu -\frac{d}{2}+r-1+\frac{2(r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle }\right) \ge {\tilde{\nu }}+O\left( \frac{1}{\langle Z\rangle }\right) . \end{aligned}$$

Using that \(k\le k_m-1\) and the interpolation bound (5.17) we may absorb the \(O\left( \frac{1}{\langle Z\rangle }\right) \) term and (5.13) is proved.

6 Pointwise bounds

We are now in position to close the control of the pointwise bounds (4.40). We start with inner bounds \(|x|\lesssim 1\):

Lemma 6.1

(Interior pointwise bounds) For all \(0\le k\le \frac{2k_m}{3}\):

$$\begin{aligned} \left| \begin{array}{l} \forall 0\le k\le \frac{2k_m}{3}, \ \ \Vert \frac{\langle Z\rangle ^{n(k)}\partial _Z^k\rho }{\rho _P}\Vert _{L^\infty (Z\le Z^*)}\le {{\mathscr {d}}}_0\\ \forall 1\le k\le \frac{2k_m}{3}, \ \ \Vert \langle Z\rangle ^{n(k)}\langle Z\rangle ^{r-2}\partial _Z^k\Psi \Vert _{L^\infty (Z \le Z^*)}\le {{\mathscr {d}}}_0 \end{array}\right. \end{aligned}$$
(6.1)

where \({{\mathscr {d}}}_0\) is a smallness constant depending on data.

Proof

We integrate (5.13) in time and obtain, by choosing \(0<{\tilde{\nu }}\ll c+c_{k_m}\), \(\forall 0\le m\le k_m-1\):

$$\begin{aligned} I_m(\tau )\le & {} e^{-2\mu {\tilde{\nu }}(\tau -\tau _0)}I_m(0)+\frac{1}{c_{k_m}-2\mu {\tilde{\nu }}}\left( e^{-2\mu {\tilde{\nu }}\tau -c_{k_m}\tau _0}-e^{-c_{k_m}\tau }\right) \nonumber \\\le & {} e^{-2\mu {\tilde{\nu }}(\tau -\tau _0)} e^{-c\tau _0}+\frac{e^{-c_{k_m}\tau _0}}{c_{k_m}-2\mu {\tilde{\nu }}}e^{-2\mu {\tilde{\nu }}\tau }\le {{\mathscr {d}}}_0 e^{-2\mu {\tilde{\nu }}\tau } \end{aligned}$$
(6.2)

for some small constant \({{\mathscr {d}}}_0\), which can be chosen to be arbitrarily small by increasing \(\tau _0\). Below, we will adjust \({{\mathscr {d}}}_0\) to remain small while absorbing any other universal constant.

Recalling (5.16):

$$\begin{aligned} \forall 0\le m\le k_m-1, \ \ \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma (m)}\le {{\mathscr {d}}}_0 e^{-\mu {\tilde{\nu }}\tau }. \end{aligned}$$
(6.3)

This, in particular, already implies bounds on the Sobolev and pointwise norms of \(({\tilde{\rho }},\Psi )\) on compact sets: for any \(Z_K<\infty \) and any \(k\le k_m-d\)

$$\begin{aligned} \Vert ({\tilde{\rho }},\Psi )\Vert _{H^k(Z\le Z_K)}\le {{\mathscr {d}}}_0 e^{-\mu {\tilde{\nu }}\tau },\quad \Vert (\partial ^k{\tilde{\rho }},\partial ^k\Psi )\Vert _{L^\infty (Z\le Z_K)}\le {{\mathscr {d}}}_0 e^{-\mu {\tilde{\nu }}\tau } \end{aligned}$$
(6.4)

case \(m\le \frac{4k_m}{9}+1=m_0\). Recall (4.29), then (6.2) implies: \(\forall 0\le m\le m_0,\)

$$\begin{aligned}&\left\| \langle Z\rangle ^{m-\frac{d}{2}+\frac{2(r-1)}{p-1}-{\tilde{\nu }}}\partial ^m_Z\rho \right\| ^2_{L^2(Z\le Z^*)}+\left\| \langle Z\rangle ^{m-\frac{d}{2}+(r-1)-{\tilde{\nu }}}\partial ^{m+1}_Z\Psi \right\| ^2_{L^2(Z\le Z^*)} \nonumber \\&\quad \le {{\mathscr {d}}}_0e^{-2\mu {\tilde{\nu }}\tau }. \end{aligned}$$
(6.5)

We now write for any spherically symmetric function u and \(\gamma >\frac{d}{2}-1\):

$$\begin{aligned} |u(Z)|\lesssim & {} |u(1)|+\int _1^Z|\partial _Zu|d\sigma \lesssim |u(1)| \nonumber \\&+\left( \int _{1\le \sigma \le Z} \frac{|\partial _Zu|^2}{\tau ^{2\gamma }}\tau ^{d-1}d\tau \right) ^{\frac{1}{2}}\left( \int _{1\le \sigma \le Z}\frac{\tau ^{2\gamma }}{\tau ^{d-1}}d\tau \right) ^{\frac{1}{2}} \nonumber \\\lesssim & {} |u(1)|+\langle Z\rangle ^{\gamma +1-\frac{d}{2}}\left\| \frac{\partial _Zu}{\langle Z\rangle ^\gamma }\right\| _{L^2(1\le \sigma \le Z)}. \end{aligned}$$
(6.6)

We pick \(1\le m\le m_0\) and apply this to \(u=Z^{\frac{2(r-1)}{p-1}}Z^{m-1}\partial _Z^{m-1}\rho \), \(\gamma +1=\frac{d}{2}+{\tilde{\nu }}\) and obtain for \(Z\le Z^*\) from (6.4) and (6.5):

$$\begin{aligned} |Z^{m-1+\frac{2(r-1)}{p-1}}\partial _Z^{m-1}\rho |\lesssim & {} e^{-c\tau }+ \langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\partial _Z(Z^{\frac{2(r-1)}{p-1}+m-1}\partial _Z^{m-1}\rho )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(2Z_2\le Z\le Z^*)}\\\lesssim & {} e^{-c\tau }+\langle Z\rangle ^{{\tilde{\nu }}}\left[ \left\| \frac{\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}}\partial _Z^m\rho }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}}}\right\| _{L^2(Z\le Z^*)} \right. \\&\left. +\left\| \frac{\langle Z\rangle ^{m-1+\frac{2(r-1)}{p-1}}\partial _Z^{m-1}\rho }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}}}\right\| _{L^2(Z\le Z^*)}\right] \\\lesssim & {} e^{-c\tau }+{{\mathscr {d}}}_0\langle Z\rangle ^{{\tilde{\nu }}}e^{-\mu {\tilde{\nu }} \tau }\le e^{-c\tau }+{{\mathscr {d}}}_0\left( \frac{\langle Z\rangle }{Z^*}\right) ^{{\tilde{\nu }}}\lesssim {{\mathscr {d}}}_0 \end{aligned}$$

and hence

$$\begin{aligned} \forall 0\le m\le \frac{4k_m}{9}, \ \ \left\| \frac{Z^m\partial _Z^m\rho }{\rho _P}\right\| _{L^\infty (Z\le Z^*)}\le {{\mathscr {d}}}_0. \end{aligned}$$

We similarly pick \(1\le m\le m_0\), apply (6.6) to \(u=\langle Z\rangle ^{r-2+m}\partial _Z^m\Psi \), \(\gamma +1=\frac{d}{2}+{\tilde{\nu }}\), and obtain for \(Z\le Z^*\) from (6.5):

$$\begin{aligned} |\langle Z\rangle ^{r-2+m}\partial _Z^m\Psi |\lesssim & {} e^{-c\tau }+\langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\partial _Z(\langle Z\rangle ^{r-2+m}\partial _Z^m\Psi )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2}\\&+ \langle Z\rangle ^{{\tilde{\nu }}} \left[ \left\| \frac{\langle Z\rangle ^{r-3+m}\partial _Z^m\Psi )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(2Z_2\le Z\le Z^*)} \right. \\&\left. +\left\| \frac{\langle Z\rangle ^{r-2+m}\partial _Z^{m+1}\Psi )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(2Z_2\le Z\le Z^*)}\right] \\\lesssim & {} e^{-c\tau }+{{\mathscr {d}}}_0\langle Z\rangle ^{{\tilde{\nu }}}e^{-\mu {\tilde{\nu }} \tau }\le e^{-c\tau }+{{\mathscr {d}}}_0\left( \frac{\langle Z\rangle }{Z^*}\right) ^{{\tilde{\nu }}}\le {{\mathscr {d}}}_0 \end{aligned}$$

and hence

$$\begin{aligned} \forall 1\le m\le m_0=\frac{4k_m}{9}+1, \ \ \Vert \langle Z\rangle ^{r-2+m}\partial _Z^m\Psi \Vert _{L^\infty (Z\le Z^*)}\le {{\mathscr {d}}}_0. \end{aligned}$$

case \(m_0\le m\le \frac{2k_m}{3}+1\). Recall (5.23):

$$\begin{aligned} \sigma (m)+m=-\alpha (k_m-m)+m=(\alpha +1)(m-m_0)+\sigma _\nu \end{aligned}$$

and rewrite the norm:

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{m,\sigma }^2\ge & {} \sum _{k=0}^m\int _{Z\le Z^*} \frac{1}{\langle Z\rangle ^{2(m-k+\sigma (m))}} \\&\times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] \\= & {} \sum _{k=0}^m\int _{Z\le Z^*} \frac{\langle Z\rangle ^{2k}}{\langle Z\rangle ^{2(\alpha +1)(m-m_0)+2\sigma _\nu }} \\&\times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}({\tilde{\rho }}^{(k)})^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi ^{(k)}|^2\right] . \end{aligned}$$

We infer, using also (6.2):

$$\begin{aligned}&\int _{2Z_2\le Z\le Z^*} \left| \frac{Z^{m-(\alpha +1)(m-m_0)}\partial _Z^{m}\rho }{\langle Z\rangle ^{\frac{d}{2}-\frac{2(r-1)}{p-1}+{\tilde{\nu }}}}\right| ^2\nonumber \\&\qquad +\int _{Z\le Z_*}\left| \frac{Z^{m-(\alpha +1)(m-m_0)}\langle Z\rangle ^{r-1}\partial ^{m+1}_Z\Psi }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}}}\right| ^2 \nonumber \\&\quad \lesssim \Vert \rho ,\Psi \Vert _{m,\sigma (m)}^2\le {{\mathscr {d}}}_0e^{-2\mu {\tilde{\nu }}\tau }. \end{aligned}$$
(6.7)

Observe that for \(m_0\le m\le \frac{2k_m}{3}+1\), from (4.30):

$$\begin{aligned}&m-(\alpha +1)(m-m_0)=m_0(1+\alpha )-\alpha m\ge m_0(1+\alpha )-\alpha \left( 2\frac{k_m}{3}+1\right) \nonumber \\&\quad = k_m\left[ \frac{4}{9}\left( 1+\frac{4}{5}\right) -\frac{2}{3}\frac{4}{5}\right] +O_{k_m\rightarrow +\infty }(1)\nonumber \\&\quad =\frac{4k_m}{15}+O_{k_m\rightarrow +\infty }(1)> \frac{k_m}{4}+10. \end{aligned}$$
(6.8)

We now apply (6.6), (6.7) to \(m_0+1\le m\le \frac{2k_m}{3}+1\), \(u=\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)-1}\partial _Z^{m-1}\rho \), \(\gamma +1=\frac{d}{2}+{\tilde{\nu }}\) and obtain for \(Z\le Z^*\):

$$\begin{aligned}&|\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)-1}\partial _Z^{m-1}\rho |\\&\quad \lesssim e^{-c\tau }+ \langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\partial _Z(\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)-1}\partial _Z^{m-1}\rho )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(Z\le Z^*)}\\&\quad \lesssim e^{-c\tau }+\langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)}\partial _Z^m\rho }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}}}\right\| _{L^2(Z\le Z^*)}\\&\qquad + \langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\langle Z\rangle ^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)-1}\partial _Z^{m-1}\rho }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}}}\right\| _{L^2(Z\le Z^*)}\\&\quad \lesssim {{\mathscr {d}}}_0\left[ 1+\langle Z\rangle ^{{\tilde{\nu }}}e^{-\mu {\tilde{\nu }}\tau }\right] \end{aligned}$$

and hence using (6.8) for \(Z\le Z^*\):

$$\begin{aligned} \left| \frac{Z^{\frac{k_m}{4}}\partial _Z^{m-1}\rho }{\rho _D}\right| \lesssim \left| Z^{m+\frac{2(r-1)}{p-1}-(\alpha +1)(m-m_0)-1}\partial _Z^{m-1}\rho \right| \lesssim {{\mathscr {d}}}_0\left[ 1+ \left( \frac{Z}{Z^*}\right) ^{{\tilde{\nu }}}\right] \lesssim {{\mathscr {d}}}_0 \end{aligned}$$

and hence

$$\begin{aligned} \forall \frac{4k_m}{9}+1\le m\le \frac{2k_m}{3}, \ \ \left\| \frac{Z^{\frac{k_m}{4}}\partial _Z^{m}\rho }{\rho _D}\right\| _{L^\infty (Z\le Z^*)}\le {{\mathscr {d}}}_0. \end{aligned}$$

For the phase, we apply (6.6), (6.7) to \(m_0+1\le m\le \frac{2k_m}{3}+1\), \(\gamma +1=\frac{d}{2}+{\tilde{\nu }}\), \(u=\langle Z\rangle ^{r-1+m-(\alpha +1)(m-m_0)}\partial _Z^m\Psi \) and obtain:

$$\begin{aligned}&\langle Z\rangle ^{r-1+m-(\alpha +1)(m-m_0)}|\partial _Z^m\Psi | \\&\quad \lesssim e^{-c\tau }+ \langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\partial _Z(\langle Z\rangle ^{r-1+m-(\alpha +1)(m-m_0)}\partial _Z^m\Psi )}{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(Z\le Z^*)}\\&\quad \lesssim e^{-c\tau }+\langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\langle Z\rangle ^{r-1+m-1-(\alpha +1)(m-m_0)}\partial _Z^m\Psi }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(Z\le Z^*)}\\&\qquad + \langle Z\rangle ^{{\tilde{\nu }}}\left\| \frac{\langle Z\rangle ^{r-1+m-(\alpha +1)(m-m_0)}\partial _Z^{m+1}\Psi }{\langle Z\rangle ^{\frac{d}{2}+{\tilde{\nu }}-1}}\right\| _{L^2(Z\le Z^*)} \\&\quad \le {{\mathscr {d}}}_0\left[ 1+\langle Z\rangle ^{{\tilde{\nu }}}e^{-\mu {\tilde{\nu }}\tau }\right] \end{aligned}$$

and hence for \(m_0+1\le m\le \frac{2k_m}{3}\) from (6.8) for \(Z\le Z^*\):

$$\begin{aligned} \langle Z\rangle ^{r-2+\frac{k_m}{4}}|\partial _Z^{m}\Psi |\lesssim \langle Z\rangle ^{r-1+m-(\alpha +1)(m-m_0)}|\partial _Z^m\Psi |\le {{\mathscr {d}}}_0, \end{aligned}$$

which concludes the proof of (6.1). \(\square \)

Similar to the above, we also have the following exterior bounds for \(|x|\ge 1\):

Lemma 6.2

(Exterior pointwise bounds) There holds:

$$\begin{aligned} \left| \begin{array}{l} \forall 0\le k\le \frac{2k_m}{3}, \ \ \Vert \frac{\langle Z\rangle ^{n(k)}\partial _Z^k\rho }{\rho _D}\Vert _{L^\infty (Z\ge Z^*)}\le {{\mathscr {d}}}_0,\\ \forall 1\le k\le \frac{2k_m}{3}, \ \ \Vert \frac{\langle Z\rangle ^{n(k)}\partial _Z^k\Psi }{b}\Vert _{L^\infty (Z \ge Z^*)}\le {{\mathscr {d}}}_0, \end{array}\right. \end{aligned}$$
(6.9)

where \({{\mathscr {d}}}_0\) is a smallness constant depending on data.

Proof

We start with the case \(0\le k\le \frac{4k_m}{9}\). We have in that case

$$\begin{aligned} I_{k,\sigma (k)}\ge & {} \int _{Z\ge Z^*} \langle Z\rangle ^{2k-2\sigma _\nu } \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}\\&\times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-1}({\tilde{\rho }}^{(k)})^2+\rho _D^2|\nabla \Psi ^{(k)}|^2\right] \end{aligned}$$

We observe from (4.9), (4.28), (5.12) and \(b={Z^*}^{2-r}\) that for \(Z\ge Z^*\)

$$\begin{aligned}&Z^{d-1}\langle Z\rangle ^{2k-2\sigma _\nu } \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}b^2|\nabla {\tilde{\rho }}^{(k)}|^2\\&\quad \approx \langle Z\rangle ^{d-1+2k-2\sigma _\nu -\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}} \frac{|\nabla {\tilde{\rho }}^{(k)}|^2}{(Z^*)^{4{\tilde{\nu }}}\rho _D^2}\\&\quad =\langle Z\rangle ^{1+2{\tilde{\nu }}} \left( \frac{\langle Z\rangle ^{k} |\nabla {\tilde{\rho }}^{(k)}|}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) ^2 \end{aligned}$$

Similarly,

$$\begin{aligned}&Z^{d-1}\langle Z\rangle ^{2k-2\sigma _\nu } \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}\rho _D^2|\nabla \Psi ^{(k)}|^2 \\&\quad \approx \langle Z\rangle ^{d-1+2k-2\sigma _\nu -\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}} \frac{|\nabla \Psi ^{(k)}|^2}{(Z^*)^{4{\tilde{\nu }}}b^2}\\&\quad = \langle Z\rangle ^{1+2{\tilde{\nu }}} \left( \frac{\langle Z\rangle ^{k} |\nabla \Psi ^{(k)}|}{(Z^*)^{2{\tilde{\nu }}}b}\right) ^2. \end{aligned}$$

Now, for a spherically symmetric function u, \(Z\ge Z^*\) and an arbitrary \(\lambda >0\)

$$\begin{aligned} |u(Z)|= & {} \left| u(Z^*)+\int _{Z^*}^Z \partial _Z u\right| \\\le & {} |u(Z^*)|+ \left( \int _{Z^*}^Z \tau ^{1+2\lambda }|\partial _Z u|^2 d\tau \right) ^{\frac{1}{2}}\left( \int _{Z^*}^Z \tau ^{-1-2\lambda }d\tau \right) ^{\frac{1}{2}} \\\le & {} |u(Z^*)|+ (Z^*)^{-\lambda }\left( \int _{Z^*}^Z \tau ^{1+2\lambda }|\partial _Z u|^2d\tau \right) ^{\frac{1}{2}}. \end{aligned}$$

We apply this to \(u=\left( \frac{\langle Z\rangle ^{k} {\tilde{\rho }}^{(k)}}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) \) for \(k\ge 1\) and \(\lambda ={\tilde{\nu }}\)

$$\begin{aligned}&\left| \left( \frac{\langle Z\rangle ^{k} {\tilde{\rho }}^{(k)}}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) (Z)\right| \le \left| \left( \frac{\langle Z\rangle ^{k} {\tilde{\rho }}^{(k)}}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) (Z^*)\right| \\&\qquad +(Z^*)^{-{\tilde{\nu }}}\left( \int _{Z^*}^Z \tau ^{1+2{\tilde{\nu }}}\left[ \left( \frac{\langle \tau \rangle ^{k} |\nabla {\tilde{\rho }}^{(k)}|}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) ^2+\left( \frac{\langle \tau \rangle ^{k-1}{\tilde{\rho }}^{(k)}}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) ^2\right] d\tau \right) ^{\frac{1}{2}}\\&\quad \lesssim (Z^*)^{-2{\tilde{\nu }}} {{\mathscr {d}}}_0+(Z^*)^{-{\tilde{\nu }}} (I_{k,\sigma (k)}+I_{k{-1},\sigma (k-1)})^{\frac{1}{2}}, \end{aligned}$$

where we used the already proved interior bounds (6.1). This, together with (6.2), immediately implies the exterior bound for \(\partial _Z^k\rho \) and \(1\le k\le \frac{4k_m}{9}+1\). The corresponding bound for \(\partial _Z^k\Psi \) is obtained similarly using \(u=\left( \frac{\langle Z\rangle ^{k} \Psi ^{(k)}}{(Z^*)^{2{\tilde{\nu }}}b}\right) \) and \(\lambda ={\tilde{\nu }}\). To prove the result for \(\rho \) in the case of \(k=0\) we note that the bootstrap assumptions imply that \({\tilde{\rho }}\rightarrow 0\), so that, together with the above estimate for \(k=1\), we have, for \(Z\ge Z_*\),

$$\begin{aligned} |{\tilde{\rho }}(Z)| = \left| \int _Z^{+\infty }\partial _Z{\tilde{\rho }}\right| \le {{\mathscr {d}}}_0\int _Z^{+\infty }\frac{\rho _D(\tau )}{\tau }d\tau \le {{\mathscr {d}}}_0\rho _D(Z) \end{aligned}$$

as desired.

Finally, we consider the regime \(\frac{4k_m}{9}+1\le k\le \frac{2k_m}{3}\). We have in that case

$$\begin{aligned} I_{k,\sigma (k)}\ge & {} \int _{Z\ge Z^*} \langle Z\rangle ^{2\alpha (k_m-k)} \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}\\&\times \left[ b^2|\nabla {\tilde{\rho }}^{(k)}|^2+(p-1)\rho _D^{p-1}({\tilde{\rho }}^{(k)})^2+\rho _D^2|\nabla \Psi ^{(k)}|^2\right] \end{aligned}$$

We observe, using \(n(k)=k_m/4\) in that range, for \(Z\ge Z^*\)

$$\begin{aligned}&Z^{d-1}\langle Z\rangle ^{2\alpha (k_m-k)} \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}b^2|\nabla {\tilde{\rho }}^{(k)}|^2 \\&\quad \approx \langle Z\rangle ^{\varpi (k)} \left( \frac{\langle Z\rangle ^{n(k)} |\nabla {\tilde{\rho }}^{(k)}|}{(Z^*)^{2{\tilde{\nu }}}\rho _D}\right) ^2 \end{aligned}$$

and

$$\begin{aligned}&Z^{d-1}\langle Z\rangle ^{2\alpha (k_m-k)} \left( \frac{Z}{Z^*}\right) ^{2n_P-\frac{4(r-1)}{p-1}-2(r-2)+4{\tilde{\nu }}}\rho _D^2|\nabla \Psi ^{(k)}|^2 \\&\quad \approx \langle Z\rangle ^{\varpi (k)} \left( \frac{\langle Z\rangle ^{k} |\nabla \Psi ^{(k)}|}{(Z^*)^{2{\tilde{\nu }}}b}\right) ^2 \end{aligned}$$

where

$$\begin{aligned} \varpi (k):= & {} 2\alpha (k_m-k)-\frac{k_m}{2}-2(r-2)+4{\tilde{\nu }}-\frac{4(r-1)}{p-1}+d-1. \end{aligned}$$

Since \(k\le \frac{2k_m}{3}\), and in view of the control of \(\alpha \) in (4.30), we have

$$\begin{aligned} \varpi (k)\ge \left( \frac{8}{15}-\frac{1}{2}\right) k_m+O(1)_{k_m\rightarrow +\infty }=\frac{1}{30}+O(1)_{k_m\rightarrow +\infty }\ge \frac{k_m}{31}. \end{aligned}$$

In particular, we have \(\varpi (k)>1\), so that the proof for \(Z\ge Z_*\) in the case \(\frac{4k_m}{9}+1\le k\le \frac{2k_m}{3}\) is analogous to the case \(0\le k\le \frac{4k_m}{9}\). Details are left to the reader. \(\square \)

7 Highest Sobolev norm

In this section we improve the bootstrap bound (4.38) on the highest unweighted Sobolev norm of \(({\tilde{\rho }},\Psi )\). Specifically, for (see (4.23))

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{{k_m}}^2=\sum _{j=0}^{k_m}\sum _{|\alpha |=j}\int \frac{b^2|\nabla \nabla ^\alpha {\tilde{\rho }}|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}(\nabla ^\alpha {\tilde{\rho }})^2+\rho _{\mathrm{Tot}}^2|\nabla \nabla ^\alpha \Psi |^2}{\langle Z\rangle ^{2(k_m-j)}}\nonumber \\ \end{aligned}$$
(7.1)

we will establish the following

Proposition 7.1

(Control of the highest Sobolev norm) For some small constant \({{\mathscr {d}}}\) dependent on the data,

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert ^2_{{k_m}}\le \Vert ({\tilde{\rho }},\Psi )(\tau _0)\Vert ^2_{{k_m}}+{{\mathscr {d}}}. \end{aligned}$$
(7.2)

Proof of Proposition 7.1

This follows from the global unweighted quasilinear energy identity. We let

$$\begin{aligned} k_m=2K_m, \ \ K_m\in {\mathbb {N}} \end{aligned}$$

and denote in this section

$$\begin{aligned} k=k_m, \ \ {\tilde{\rho }}^{(k)}=\Delta ^{K_m}{\tilde{\rho }}, \ \ \Psi ^{(k)}=\Delta ^{K_m}\Psi . \end{aligned}$$

We recall the notation (5.11)

$$\begin{aligned} I_{k_m}=\int b^2|\nabla \partial ^{k_m}{\tilde{\rho }}|^2+(p-1)\int \rho _D^{p-2}\rho _{\mathrm{Tot}} |\partial ^{k_m}{\tilde{\rho }}|^2+\int \rho _{\mathrm{Tot}}^2|\nabla \partial ^{k_m}\Psi |^2.\nonumber \\ \end{aligned}$$
(7.3)

Step 1 Control of lower order terms. We recall the notation:

$$\begin{aligned} \left| \begin{array}{ll} \Vert {\tilde{\rho }},\Psi \Vert _{k_m,\sigma ({m})}^2=\sum _{j=0}^{k_m}\int \chi _{j,k_m,\sigma ({m})}b^2|\nabla \partial ^j{\tilde{\rho }}|^2\\ \quad +(p-1)\int \chi _{j,k_m}\rho _D^{p-2}\rho _{\mathrm{Tot}}(\partial ^j{\tilde{\rho }})^2 +\int \chi _{j,k_m}\rho _{\mathrm{Tot}}^2|\nabla \partial ^j\Psi |^2\\ \chi _{j,k_m,\sigma ({m})}(Z)=\frac{1}{\langle Z\rangle ^{2(k_m-j)}}. \end{array}\right. \end{aligned}$$

In view of (5.16) and (6.2), we have

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{k_m,\sigma ({m})}^2\le \sum _{k=0}^{k_m}I_{k}\le I_{k_m}+{{\mathscr {d}}}_{0}. \end{aligned}$$
(7.4)

By Remark 4.2 we can replace (up to the lower order terms controlled as above) \(I_{k_m}\) with

$$\begin{aligned} J_{k_m}:= & {} \int b^2|\nabla \Delta ^{K_m}{\tilde{\rho }}|^2+(p-1)\int \rho _D^{p-2}\rho _{\mathrm{Tot}} |\Delta ^{K_m}{\tilde{\rho }}|^2\nonumber \\&+\int \rho _{\mathrm{Tot}}^2|\nabla \Delta ^{K_m}\Psi |^2. \end{aligned}$$
(7.5)

We claim: there exist \(k_m^*(d,r,p),c_{d,r,p}>0\) such that for all \(k_m>k_m^*(d,r,p)\), there holds:

$$\begin{aligned} \frac{d}{d\tau }\left\{ J_{k_m}\left[ 1+O(\delta )\right] \right\} +c_{d,r}k_mJ_{k_m}\le {{\mathscr {d}}}. \end{aligned}$$
(7.6)

Integrating the above in time, using (4.24), (7.4), yields (7.2).

Step 2 Energy identity. We revisit the computation of (5.6), (5.7), (5.8), (5.3) in order to extract all the coupling terms at the highest level of derivatives. Recall (5.3):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau {\tilde{\rho }}=-\rho _{\mathrm{Tot}}\Delta \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi +H_1{\tilde{\rho }}-H_2\Lambda {\tilde{\rho }}-\tilde{{\mathcal {E}}}_{P,\rho }\\ \partial _\tau \Psi =b^2\frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}-\left\{ H_2\Lambda \Psi +\mu (r-2)\Psi \right. \\ \quad \qquad \left. +|\nabla \Psi |^2+(p-1)\rho _D^{p-2}{\tilde{\rho }}+\text {NL}({\tilde{\rho }})\right\} -\tilde{{\mathcal {E}}}_{P,\Psi }. \end{array}\right. \end{aligned}$$

We use

$$\begin{aligned}{}[\Delta ^{K_m},\Lambda ]=k_m\Delta ^{K_m} \end{aligned}$$

and recall (C.1):

$$\begin{aligned}{}[\Delta ^k,V]\Phi -2k\nabla V\cdot \nabla \Delta ^{k-1}\Phi =\sum _{|\alpha |+|\beta |=2k,|\beta |\le 2k-2}c_{k,\alpha ,\beta }\partial ^\alpha V\partial ^\beta \Phi , \end{aligned}$$

which gives

$$\begin{aligned} \Delta ^{K_m}(H_2\Lambda {\tilde{\rho }})=k_m(H_2+\Lambda H_2)\rho _k+H_2\Lambda \rho _k+{\mathcal {A}}_k({\tilde{\rho }}) \end{aligned}$$

with

$$\begin{aligned} \left| \begin{array}{l} |{\mathcal {A}}_k({\tilde{\rho }})|\lesssim c_k\sum _{j=1}^{k-1}\frac{|\nabla ^j{\tilde{\rho }}|}{\langle Z\rangle ^{k_m+r-j}},\\ |\nabla {\mathcal {A}}_k({\tilde{\rho }})|\lesssim c_k\sum _{j=1}^{k}\frac{|\nabla ^j{\tilde{\rho }}|}{\langle Z\rangle ^{k_m+r+1-j}}, \end{array}\right. \end{aligned}$$
(7.7)

where \(\nabla ^j=\partial ^{\alpha _1}_1\cdots \partial ^{\alpha _d}_d\), \(j=\alpha _1+\cdots +\alpha _d\) denotes a generic derivative of order j. Using (C.1) again:

$$\begin{aligned} \partial _\tau {\tilde{\rho }}^{(k)}= & {} (H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k-H_2\Lambda {\tilde{\rho }}_k-(\Delta ^{K_m}\rho _{\mathrm{Tot}})\Delta \Psi \nonumber \\&-k\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi ^{(k)}-\rho _{\mathrm{Tot}}\Delta \Psi _k\nonumber \\&- 2\nabla (\Delta ^{K_m}\rho _{\mathrm{Tot}})\cdot \nabla \Psi -2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+ F_1 \end{aligned}$$
(7.8)

with

$$\begin{aligned} F_1= & {} -\Delta ^{K_m}\tilde{{\mathcal {E}}}_{P,\rho }+[\Delta ^{K_m},H_1]{\tilde{\rho }}-A_k({\tilde{\rho }})\nonumber \\&- \sum _{\left| \begin{array}{ll} j_1+j_2=k\\ j_1\ge 2, j_2\ge 1\end{array}\right. }c_{j_1,j_2}\nabla ^{j_1}\rho _{\mathrm{Tot}}\nabla ^{j_2}\Delta \Psi \nonumber \\&-\sum _{\left| \begin{array}{ll}j_1+j_2=k\\ j_1,j_2\ge 1\end{array}\right. }c_{j_1,j_2}\nabla ^{j_1}\nabla \rho _{\mathrm{Tot}}\cdot \nabla ^{j_2}\nabla \Psi . \end{aligned}$$
(7.9)

For the second equation, we have similarly:

$$\begin{aligned} \partial _\tau \Psi _k= & {} b^2\left( \frac{\Delta ^{K_m+1} \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}} -\frac{k\nabla \Delta ^{K_m}\rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right) \nonumber \\&- k(H_2+\Lambda H_2)\Psi _k-H_2\Lambda \Psi _k-\mu (r-2)\Psi _k-2\nabla \Psi \cdot \nabla \Psi _k\nonumber \\&- \left[ (p-1)\rho _P^{p-2}{\tilde{\rho }}_k+k(p-1)(p-2)\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\right] \nonumber \\&+F_2 \end{aligned}$$
(7.10)

with

$$\begin{aligned} F_2= & {} -\partial ^k\tilde{{\mathcal {E}}}_{P,\Psi }+b^2\left[ \Delta ^{K_m}\left( \frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right) -\frac{\Delta ^{K_m+1} \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}} +\frac{k\nabla \Delta ^{K_m}\rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right] \nonumber \\&-(p-1)\left( [\Delta ^{K_m},\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\right) \nonumber \\&-{\mathcal {A}}_k(\Psi ) - \sum _{j_1+j_2=k,j_1,j_2\ge 1}\nabla ^{j_1}\nabla \Psi \cdot \nabla ^{j_2}\nabla \Psi -\Delta ^{K_m}\text {NL}({\tilde{\rho }}) \end{aligned}$$
(7.11)

and

$$\begin{aligned} \left| \begin{array}{l} |{{\mathcal {A}}}_k(\Psi )|\lesssim \sum _{j=1}^{k-1}\frac{|\nabla ^j\Psi |}{\langle Z\rangle ^{k_m+r-j}}\\ |\nabla {\mathcal {A}}_k(\Psi )|\lesssim \sum _{j=1}^{k}\frac{|\nabla ^j\Psi |}{\langle Z\rangle ^{k_m+r+1-j}}. \end{array}\right. \end{aligned}$$
(7.12)

We then run the global quasilinear energy identity similar to (5.10) with \(\chi =1\) and obtain:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int b^2|\nabla {\tilde{\rho }}_k|^2+(p-1)\int \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2+\int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right\} \nonumber \\&\quad = -\mu (r-2)b^2\int |\nabla {\tilde{\rho }}_k|^2+\int \partial _\tau \rho _{\mathrm{Tot}}\left[ \frac{p-1}{2}\rho _D^{p-2}{\tilde{\rho }}_k^2+\rho _{\mathrm{Tot}}|\nabla \Psi _k|^2\right] \nonumber \\&\qquad + \frac{p-1}{2}\int (p-2)\partial _\tau \rho _D\rho _D^{p-3}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2\nonumber \\&\qquad + \int F_1(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k+b^2\int \nabla F_1\cdot \nabla {\tilde{\rho }}_k+\int \rho ^2_T\nabla F_2\cdot \nabla \Psi _k\nonumber \\&\qquad - \int k\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k(-b^2\Delta {\tilde{\rho }}_k+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k)\nonumber \\&\qquad +\int b^2\frac{k\nabla \Delta ^{K_m}\rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}(\rho _{\mathrm{Tot}}\Delta \Psi _k+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k)\nonumber \\&\qquad + \int \left[ (H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k-H_2\Lambda {\tilde{\rho }}_k\right. \nonumber \\&\qquad \left. -(\Delta ^{K_m}\rho _{\mathrm{Tot}})\Delta \Psi -2\nabla (\Delta ^{K_m}\rho _{\mathrm{Tot}})\cdot \nabla \Psi \right] \nonumber \\&\qquad \times \left[ -b^2\Delta {\tilde{\rho }}_k+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k\right] \nonumber \\&\qquad - \int \left[ b^2\rho _{\mathrm{Tot}}\Delta ^{K_m+1} \rho _D-k\rho _{\mathrm{Tot}}(H_2+\Lambda H_2)\Psi _k\right. \nonumber \\&\qquad \left. -\rho _{\mathrm{Tot}} H_2\Lambda \Psi _k-\mu (r-2)\rho _{\mathrm{Tot}}\Psi _k-2\rho _{\mathrm{Tot}}\nabla \Psi \cdot \nabla \Psi _k\right] \nonumber \\&\qquad \times \left[ 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right] + \int k(p-1)(p-2)\rho _{\mathrm{Tot}}\rho _D^{p-3}\nabla \rho _D\nonumber \\&\qquad \cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\left[ 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right] . \end{aligned}$$
(7.13)

We now estimate all terms in (7.13). The proof is similar to that one of Proposition 5.2 with two main differences: the absence of a cut-off function \(\chi \), and a priori control of lower order derivatives from (7.4). The challenge here is to avoid any loss of derivatives and to compute exactly the quadratic form at the highest level of derivatives. The latter will be shown to be positive on a compact set in Z provided \(k_m>k_m^*(d,r,p)\gg 1\) has been chosen large enough.

In what follows, below, we will use \(\delta >0\) as a small universal constant and will assume that the pointwise bounds (6.1) obtained on the lower order derivatives of \({\tilde{\rho }}\) and \(\Psi \) are dominated by \(\delta \). On the set \(Z\le Z^*\), this will often be a source of smallness, while for \(Z\ge Z^*\), we may use the bootstrap bounds (4.40) and the \(\delta \)-smallness will be generated by extra powers of Z. We also note that from (7.6) the quadratic form is expected to be proportionate to \(k_m I_{k_m}\). Choosing \(k_m\) large will allow us to dominate other quadratic terms without smallness but with the uniform dependence on \(k_m\). The notation \(\lesssim \) will allow dependence on \(k_m\), while O will indicate a bound independent of \(k_m\). As before, \({{\mathscr {d}}}_0\) (as well as \({{\mathscr {d}}}\)) will denote small constants, dependent on the data (or, more precisely, on \(\tau _0\)), that can be made arbitrarily small. In particular, we will use

$$\begin{aligned} \Vert {\tilde{\rho }},\Psi \Vert _{k_m-1,\sigma (k_m-1)}\le {{\mathscr {d}}}_0. \end{aligned}$$
(7.14)

The constants \(\delta \gg {{\mathscr {d}}}_0\) will be assumed to be smaller than any power of \(k_m\), so that our calculations will be unaffected by combinatorics generated by taking \(k_m\) derivatives of the equations.

Step 3 Leading order terms.

Cross term. Recall (5.27):

$$\begin{aligned} \int \Delta g F\cdot \nabla g\,dx= & {} \sum _{i,j=1}^d \int \partial _i^2 g F_j\partial _jg\,dx\\= & {} -\sum _{i,j=1}^d \int \partial _ig(\partial _iF_j\partial _jg+F_j\partial ^2_{i,j}g) \\= & {} -\sum _{i,j=1}^d \int \partial _iF_j\partial _ig\partial _jg+\frac{1}{2}\int |\nabla g|^2\nabla \cdot F. \end{aligned}$$

Letting \(g=g_1+g_2\) yields a bilinear off-diagonal Pohozhaev identity:

$$\begin{aligned}&\int \left[ \Delta g_1 F\cdot \nabla g_2+\Delta g_2 F\cdot \nabla g_1\right] dx \\&\quad =-\sum _{i,j=1}^d \int \partial _iF_j(\partial _ig_1\partial _jg_2+\partial _ig_2\partial _jg_1) \\&\qquad +\int \nabla g_1\cdot \nabla g_2(\nabla \cdot F). \end{aligned}$$

We may therefore integrate by parts the one term in (7.13) which has too many derivatives:

$$\begin{aligned}&b^2k\left| \int \left[ \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k\Delta {\tilde{\rho }}_k+\nabla \Delta ^{K_m}\rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}\Delta \Psi _k\right] \right| \\&\quad = b^2k\left| \int \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k\Delta {\tilde{\rho }}_k+\nabla \rho _{\mathrm{Tot}}\cdot \nabla {\tilde{\rho }}_k\Delta \Psi _k+\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Delta ^{K_m}\rho _D\Delta \Psi _k\right| \\&\quad = b^2k\Bigg |-\int \sum _{i,j=1}^d\partial ^2_{i,j}\rho _{\mathrm{Tot}}(\partial _i{\tilde{\rho }}_k\partial _j\Psi _k+\partial _i\Psi _k\partial _j{\tilde{\rho }}_k)+\int \nabla {\tilde{\rho }}_k\cdot \nabla \Psi _k\Delta \rho _{\mathrm{Tot}}\\&\qquad -\int \nabla \Psi _k\cdot \nabla (\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Delta ^{K_m}\rho _D)\Bigg |\\&\quad \lesssim b^2k\int \rho _{\mathrm{Tot}}|\nabla \Psi _k|\left[ \frac{1}{\langle Z\rangle ^{k+2}}+\frac{|\nabla {\tilde{\rho }}_k|}{\langle Z\rangle ^2}\right] \\&\quad \le \delta \int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2+C_\delta b^4+C_\delta b^4\int |\nabla {\tilde{\rho }}_k|^2\\&\quad \lesssim \delta J_{k_m}+C_\delta b^4. \end{aligned}$$

We estimate similarly:

$$\begin{aligned}&\left| kb^2\int \frac{\nabla \Delta ^{K_m}\rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k\right| \\&\quad \le \delta \left[ b^2\int |\nabla {\tilde{\rho }}_k|^2+\int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right] +c_\delta b^4\lesssim \delta J_{k_m}+C_\delta b^4. \end{aligned}$$

We use

$$\begin{aligned} \frac{|{\tilde{\rho }}|}{\rho _{\mathrm{Tot}}}+\frac{|\Lambda {\tilde{\rho }}|}{\langle Z\rangle ^c\rho _{\mathrm{Tot}}}\lesssim \delta , \ \ 0<c\ll 1 \end{aligned}$$
(7.15)

to compute the first coupling term:

$$\begin{aligned}&-k(p-1)\int \nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k \\&\quad = -k\int \rho _D\nabla \rho ^{p-1}_D\cdot \nabla \Psi _k{\tilde{\rho }}_k+ O\left( \delta \int \frac{|\nabla \Psi _k|\rho _D^{p-1}\rho _{\mathrm{Tot}}|{\tilde{\rho }}_k|}{\langle Z\rangle }\right) \\&\quad = -k\int \rho _D\nabla \rho _D^{p-1}\cdot \nabla \Psi _k{\tilde{\rho }}_k+O\left( \delta J_{k_m}\right) . \end{aligned}$$

The second coupling term is computed after an integration by parts using (7.15), the control of lower order terms (7.4) and the spherically symmetric assumption:

$$\begin{aligned}&\int (\rho _{\mathrm{Tot}}\Delta \Psi _k+2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k)k(p-1)(p-2)\rho _{\mathrm{Tot}}\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\\&\quad = k(p-1)(p-2)\int \nabla \cdot (\rho _{\mathrm{Tot}}^2\nabla \Psi _k)\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\\&\quad = -k(p-1)(p-2)\int \rho _{\mathrm{Tot}}^2\nabla \Psi _k\cdot \nabla \left( \rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\right) \\&\quad = -k(p-1)(p-2)\int \rho _{\mathrm{Tot}}^2\partial _Z\Psi _k\partial _Z\left( \rho _D^{p-3}\partial _Z\rho _D\partial _Z\Delta ^{K_m-1}{\tilde{\rho }}\right) \\&\quad = -k(p-1)(p-2)\int \rho _D^{p-3}\partial _Z\rho _D\rho _{\mathrm{Tot}}^2\partial _Z\Psi _k\partial ^2_Z\Delta ^{K_m-1}{\tilde{\rho }}\\&\qquad +O\left( \int \rho _{\mathrm{Tot}}|\nabla \Psi _k|\rho _{\mathrm{Tot}}^{p-1}\frac{|\nabla ^{k_m-1}{\tilde{\rho }}|}{\langle Z\rangle ^2}\right) \\&\quad =-\int k(p-2)\rho _D\partial _Z(\rho _D^{p-1})\partial _Z\Psi _k {\tilde{\rho }}_k\\&\qquad +\int k(p-2)(d-1)\rho _D\partial _Z(\rho _D^{p-1})\partial _Z\Psi _k\frac{\partial _Z\Delta ^{K_m-1}{\tilde{\rho }}}{|Z|}\\&\qquad +O\left( \int \rho _{\mathrm{Tot}}|\nabla \Psi _k|\rho _{\mathrm{Tot}}^{p-1}\frac{|\nabla ^{k_m-1}{\tilde{\rho }}|}{\langle Z\rangle ^2}\right) \\&\quad = -k(p-2)\int \rho _D\nabla \rho ^{p-1}_D\cdot \nabla \Psi _k{\tilde{\rho }}_k+O\left( {{\mathscr {d}}}_0+\delta J_{k_m}\right) , \end{aligned}$$

where in the last step we used that \( \frac{|\partial _Z \rho _D|}{|Z|\rho _D} \lesssim \frac{1}{\langle Z\rangle ^2}. \)

\(\rho _k\) terms. We compute:

$$\begin{aligned}&\int (H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k(-b^2\Delta {\tilde{\rho }}_k+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k)\\&\quad = \int (H_1-k(H_2+\Lambda H_2))\left[ b^2|\nabla {\tilde{\rho }}_k|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right] \\&\qquad -\frac{b^2}{2}\int {\tilde{\rho }}_k^2\Delta (H_1-k(H_2+\Lambda H_2)). \end{aligned}$$

We now use the global lower bound, see properties (2.21) and (2.22) of the the profile \((w,\sigma )\),

$$\begin{aligned} H_2+\Lambda H_2=\mu (1-w-\Lambda w)\ge c_{p,d,r}, \ \ c_{p,d,r}>0 \end{aligned}$$

to estimate using (8.17), (7.14):

$$\begin{aligned}&\int (H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k(-b^2\Delta {\tilde{\rho }}_k+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k)\\&\quad \le -k\int \left[ 1+O_{k_m\rightarrow +\infty }\left( \frac{1}{k_m}\right) \right] (H_2+\Lambda H_2)\\&\qquad \times \left[ b^2|\nabla {\tilde{\rho }}_k|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right] +Cb^2\int \frac{{\tilde{\rho }}_k^2}{\langle Z\rangle ^{2+r}}\\&\quad \le -k\int (H_2+\Lambda H_2)\left[ b^2|\nabla {\tilde{\rho }}_k|^2+(p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right] +{{\mathscr {d}}}_0. \end{aligned}$$

Next, using

$$\begin{aligned} |\partial ^k\rho _D|\lesssim \frac{\rho _D}{\langle Z\rangle ^{k}}, \end{aligned}$$

we estimate from (4.40):

$$\begin{aligned}&b^2\left| \int \Delta {\tilde{\rho }}_k\left[ (\Delta ^{K_m}\rho _D)\Delta \Psi +2\nabla (\Delta ^{K_m}\rho _D)\cdot \nabla \Psi \right] \right| \\&\quad \lesssim b^2\int |\nabla {\tilde{\rho }}_k|\left[ |\nabla (\Delta ^{K_m}\rho _D\Delta \Psi )|+|\nabla (\nabla \Delta ^{K_m}\rho _D\cdot \nabla \Psi )|\right] \\&\quad \le b^2\delta \int |\nabla {\tilde{\rho }}_k|^2+\frac{b^2}{\delta }\sum _{j=1}^3\int \frac{\rho _D^2|\partial ^j\Psi |^2}{\langle Z\rangle ^{2\left( k+3-j\right) }}\le \delta J_{k_m}+{{\mathscr {d}}}_0 b^2. \end{aligned}$$

For the nonlinear term, we use (6.1), (4.40), (5.27), (7.4) to estimate

$$\begin{aligned}&b^2\left| \int \Delta {\tilde{\rho }}_k\left[ {\tilde{\rho }}_k\Delta \Psi +2\nabla {\tilde{\rho }}_k\cdot \nabla \Psi \right] \right| \\&\quad \lesssim b^2\left[ \int |\partial ^2\Psi ||\nabla {\tilde{\rho }}_k|^2+\int \frac{{\tilde{\rho }}_k^2}{\langle Z\rangle ^{2}}\right] \le \delta J_{k_m}+{{\mathscr {d}}}_0 b^2. \end{aligned}$$

Next

$$\begin{aligned}&\left| \int \left[ (\Delta ^{K_m}\rho _D)\Delta \Psi -2\nabla (\Delta ^{K_m}\rho _D)\cdot \nabla \Psi \right] (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k\right| \\&\quad \le \delta \int \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2+\frac{C}{\delta }\int \rho _D^{p-2}\rho _{\mathrm{Tot}}^2\left[ \frac{|\partial ^2\Psi |^2}{\langle Z\rangle ^{{2 k}}}+\frac{|\partial \Psi |^2}{\langle Z\rangle ^{2(k+1)}}\right] \\&\quad \le \delta J_{k_m}+{{\mathscr {d}}}_0, \end{aligned}$$

since we are assuming that \({{\mathscr {d}}}_0\ll \delta \), and for the nonlinear term after an integration by parts:

$$\begin{aligned} \left| \int \left[ {\tilde{\rho }}_k\Delta \Psi -2\nabla {\tilde{\rho }}_k\cdot \nabla \Psi \right] (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k\right| \lesssim \delta \int \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2. \end{aligned}$$

From Pohozhaev (5.27):

$$\begin{aligned} -\int H_2\Lambda {\tilde{\rho }}_k(-b^2\Delta {\tilde{\rho }}_k)=b^2\int \Delta {\tilde{\rho }}_k(ZH_2)\cdot \nabla {\tilde{\rho }}_k= O\left( b^2\int |\nabla {\tilde{\rho }}_k|^2\right) . \end{aligned}$$

Integrating by parts and using (8.17), (5.41), (5.42):

$$\begin{aligned}&-\int H_2\Lambda {\tilde{\rho }}_k\left[ (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k\right] \\&\qquad +\frac{p-1}{2}\int (p-2)\partial _\tau \rho _D\rho _D^{p-3}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2+\frac{p-1}{2}\int \partial _\tau \rho _{\mathrm{Tot}}\rho ^{p-2}_D{\tilde{\rho }}_k^2\\&\quad = \frac{p-1}{2}\int {\tilde{\rho }}_k^2\left[ \nabla \cdot (ZH_2\rho _D^{p-2}\rho _{\mathrm{Tot}})+\partial _\tau (\rho _D^{p-2})\rho _{\mathrm{Tot}}+ \partial _\tau \rho _{\mathrm{Tot}}\rho ^{p-2}_D\right] \\&\quad = O\left( \int \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k^2\right) \end{aligned}$$

Note that the above two bounds, even though dependent on the highest order derivatives, contain no k dependence.

\(\Psi _k\) terms. After an integration by parts:

$$\begin{aligned}&\left| \int b^2\Delta ^{K_m+1}\rho _D\left[ 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right] \right| \\&\quad \lesssim b^2\int \rho _{\mathrm{Tot}}\frac{|\nabla \Psi _k|}{\langle Z\rangle ^{k+3}}\le \delta \int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2+{{\mathscr {d}}}_0. \end{aligned}$$

Then

$$\begin{aligned}&\mu (r-2)\int \rho _{\mathrm{Tot}}\Psi _k\left[ 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right] \\&\quad = -\mu (r-2)\int \Psi _k^2\nabla \cdot (\rho _{\mathrm{Tot}}\nabla \rho _{\mathrm{Tot}})-\mu (r-2)\int \nabla \Psi _k\cdot \nabla (\rho _{\mathrm{Tot}}^2\Psi _k)\\&\quad = -\mu (r-2)\int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2 \end{aligned}$$

and similarly, using (8.17), (7.4):

$$\begin{aligned}&k\int \rho _{\mathrm{Tot}}(H_2+\Lambda H_2)\Psi _k\left[ 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right] \\&\quad =k\int (H_2+\Lambda H_2)\Psi _k\nabla \cdot (\rho _{\mathrm{Tot}}^2\nabla \Psi _k)\\&\quad = -k\left[ \int (H_2+\Lambda H_2)\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right. \\&\qquad \left. +\int \rho _{\mathrm{Tot}}^2\Psi _k^2\left( \frac{\nabla \cdot (\rho _{\mathrm{Tot}}^2\nabla (H_2+\Lambda H_2))}{2\rho ^2_T}\right) \right] \\&\quad = -k\int (H_2+\Lambda H_2)\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2+{{\mathscr {d}}}_0, \end{aligned}$$

where the \(\Psi _k^2\) term is controlled, with the help of the bound

$$\begin{aligned} \left| \frac{\nabla \cdot (\rho _{\mathrm{Tot}}^2\nabla (H_2+\Lambda H_2))}{2\rho ^2_T}\right| \lesssim \langle Z\rangle ^{-2-r}, \end{aligned}$$

by using the already bounded \(\Vert ({\tilde{\rho }},\Psi )\Vert _{k_m-1,\sigma (k_m-1)}\)-norm.

Then, from (6.1) and (6.9):

$$\begin{aligned} \left| \int 2\rho _{\mathrm{Tot}}\nabla \Psi \cdot \nabla \Psi _k(2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k)\right| \lesssim {{\mathscr {d}}}_0\int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2 \end{aligned}$$

and using (5.27):

$$\begin{aligned} \left| \int 2\rho _{\mathrm{Tot}}\nabla \Psi \cdot \nabla \Psi _k(\rho _{\mathrm{Tot}}\Delta \Psi _k)\right|\lesssim & {} \int |\nabla \Psi _k|^2||\partial (\rho _{\mathrm{Tot}}^2\nabla \Psi )|\\\lesssim & {} {{\mathscr {d}}}_0\int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2. \end{aligned}$$

Arguing verbatim as in the proof of (5.49) produces the bound

$$\begin{aligned}\left| \int \rho _{\mathrm{Tot}}H_2\Lambda \Psi _k\left( 2\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+\rho _{\mathrm{Tot}}\Delta \Psi _k\right) \right| =O\left( \int \rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right) . \end{aligned}$$

Step 4 \(F_1\) terms. We claim the bound:

$$\begin{aligned} b^2\int |\nabla F_1|^2+(p-1)\int {\rho }_{D}^{p-1}F_1^2\lesssim \delta J_{k_m}+ {{\mathscr {d}}}_0. \end{aligned}$$
(7.16)

Source term induced by localization. From (5.55), for \(k_m\) large enough:

$$\begin{aligned} \int \rho _D^{p-2}\rho _{\mathrm{Tot}}|\Delta ^{K_m}\tilde{{\mathcal {E}}}_{P,\rho }|^2+b^2\int |\nabla \Delta ^{K_m}\tilde{{\mathcal {E}}}_{P,\rho }|^2\lesssim {{\mathscr {d}}}_0. \end{aligned}$$

\([\Delta ^{K_m},H_1]\) term. We estimate from (5.56), (7.14)

$$\begin{aligned} (p-1)\int \rho _D ^{p-1}([\Delta ^{K_m},H_1]{\tilde{\rho }})^2\lesssim \sum _{j=0}^{k-1}\int \rho _D^{p-1}\frac{|\partial ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\le {{\mathscr {d}}}_0 \end{aligned}$$

and

$$\begin{aligned}&b^2\int |\nabla ([\Delta ^{K_m},H_1]{\tilde{\rho }})|^2\lesssim b^2 \sum _{j=0}^{k}\int \frac{|\partial _j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(1+r+k-j)}}\\&\quad = b^2\int \frac{{\tilde{\rho }}^2dZ}{\langle Z\rangle ^{2(1+r+k)}}+b^2 \sum _{j=0}^{k-1}\int \frac{|\partial ^j\nabla {\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k+1-j)+2}}\\&\quad \lesssim b^2+\Vert {\tilde{\rho }},\Psi \Vert _{k_m-1,\sigma (k_m-1)}\le {{\mathscr {d}}}_0. \end{aligned}$$

\({\mathcal {A}}_k({\tilde{\rho }})\) term. From (7.7), (7.14):

$$\begin{aligned} (p-1)\int \rho _D^{p-1}({\mathcal {A}}_k({\tilde{\rho }}))^2\lesssim \sum _{j=1}^{k-1}\int \rho _D^{p-1} \frac{|\nabla ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\le {{\mathscr {d}}}_0 \end{aligned}$$

and

$$\begin{aligned} b^2\int |\nabla ({\mathcal {A}}_k({\tilde{\rho }}))|^2\lesssim b^2 \sum _{j=0}^{k-1}\int \frac{|\nabla \nabla ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(r+k-j)}}\le {{\mathscr {d}}}_0 \end{aligned}$$

and (7.16) is proved for this term.

Nonlinear term. After changing indices, we need to estimate

$$\begin{aligned} N_{j_1,j_2}=\nabla ^{j_1}\rho _{\mathrm{Tot}}\nabla ^{j_2}\nabla \Psi , \ \ j_1+j_2=k+1, \ \ 2\le j_1,j_2\le k-1. \end{aligned}$$

For the profile term:

$$\begin{aligned} |\partial ^{j_1}\rho _D\nabla ^{j_2}\nabla \Psi |\lesssim \rho _{D}\frac{|\nabla ^{j_2}\nabla \Psi |}{\langle Z\rangle ^{j_1}}= \rho _D\frac{|\nabla ^{j_2}\nabla \Psi |}{\langle Z\rangle ^{k+1-j_2}} \end{aligned}$$

and therefore, recalling (5.59), (7.14):

$$\begin{aligned} \int (p-1) N_{j_1,j_2}^2\rho _D^{p-1}\lesssim \int \frac{\rho _{\mathrm{Tot}}^2|\nabla ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)+2(r-1)}}\le {{\mathscr {d}}}_0. \end{aligned}$$

Similarly, after taking a derivative:

$$\begin{aligned} b^2\int |\nabla N_{j_1,j_2}|^2\lesssim & {} b^2\int \frac{\rho _{\mathrm{Tot}}^2|\nabla ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+2-j_2)}} +b^2\int \frac{\rho _{\mathrm{Tot}}^2|\nabla ^{j_2+1}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)}}\\\le & {} {{\mathscr {d}}}_0+\delta J_{k_m}. \end{aligned}$$

The \(\delta J_{k_m}\) term above controls the case \(j_2=k-1\).

We now turn to the control of the nonlinear term. If \(j_1\le \frac{4k_m}{9}\), then from (4.40), (7.4):

$$\begin{aligned}&\int \rho _D^{p-1}|\nabla ^{j_1}{\tilde{\rho }}\nabla ^{j_2}\nabla \Psi |^2\lesssim \int \rho _D ^2\frac{|\nabla ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)+(p-1)\frac{2(r-1)}{p-1}}}\le {{\mathscr {d}}}_0. \end{aligned}$$

If \(j_2\le \frac{4k_m}{9}\), then from (4.40) with \(b=\frac{1}{(Z^*)^{r-2}}\):

$$\begin{aligned} \int \rho _D^{p-1}|\nabla ^{j_1}{\tilde{\rho }}\nabla ^{j_2}\nabla \Psi |^2\lesssim & {} \int _{Z\le Z^*} \rho _D^{p-1}\frac{|\nabla ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k+1+(r-2)-j_1)}}\\&+b^2\int _{Z\ge Z^*}\rho _D^{p-1}\frac{|\nabla ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k+1-j_1)}}\le {{\mathscr {d}}}_0. \end{aligned}$$

We may therefore assume \(j_1,j_2\ge m_0=\frac{4k_m}{9}+1\), which implies \(k\ge m_0\) and \(j_1,j_2\le \frac{2k_m}{3}\) and hence from (4.40) and (6.1):

$$\begin{aligned} \int \rho _D^{p-1}|\partial ^{j_1}{\tilde{\rho }}\nabla ^{j_2}\nabla \Psi |^2\lesssim {{\mathscr {d}}}_0+\int _{Z\ge Z^*}\frac{dZ}{\langle Z\rangle ^{\frac{k_m}{10}}}\le {{\mathscr {d}}}_0. \end{aligned}$$

The \(b^2\) derivative contribution of the nonlinear term is estimated similarly.

Step 5 \(F_2\) terms. We claim:

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla (F_2+\Delta ^{K_m}\text {NL}({\tilde{\rho }}))|^2 \le \delta J_{k_m}+ {{\mathscr {d}}}_0. \end{aligned}$$
(7.17)

The nonlinear term \(\Delta ^{K_m}\text {NL}({\tilde{\rho }})\) will be treated in the next step.

\({\mathcal {A}}_k(\Psi )\) term. From (7.12)

$$\begin{aligned} |\nabla {\mathcal {A}}_k(\Psi )|\lesssim \sum _{j=1}^{k}\frac{|\nabla ^j\Psi |}{\langle Z\rangle ^{r+k-j+1}} \end{aligned}$$

and hence:

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla {\mathcal {A}}_k(\Psi )|^2\lesssim \sum _{j=0}^{k-1}\int \rho _{\mathrm{Tot}}^2\frac{|\nabla \nabla ^j\Psi |^2}{\langle Z\rangle ^{2(r+k-j)}}\le {{\mathscr {d}}}_0. \end{aligned}$$

\([\Delta ^{K_m},\rho _D^{p-2}]\) term. From (C.1):

$$\begin{aligned} \left| [\Delta ^{K_m},\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\right| \lesssim \sum _{j=0}^{k-2}\frac{|\nabla ^j{\tilde{\rho }}|}{\langle Z\rangle ^{k-j}}\rho _D^{p-2}. \end{aligned}$$

After taking a derivative:

$$\begin{aligned}&\int \rho _{\mathrm{Tot}}^2\left| \nabla \left[ [\Delta ^{K_m},\rho _D^{p-2}]{\tilde{\rho }}-k(p-2)\rho _D^{p-3}\nabla \rho _D\cdot \nabla \Delta ^{K_m-1}{\tilde{\rho }}\right] \right| ^2\\&\quad \lesssim \sum _{j=0}^{k-1}\int \rho _D^{2(p-2)+2}\frac{|\nabla ^j{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-j)+2}}\le {{\mathscr {d}}}_0. \end{aligned}$$

Nonlinear \(\Psi \) term. Let

$$\begin{aligned} \partial N_{j_1,j_2}=\nabla ^{j_1}\nabla \Psi \nabla ^{j_2}\nabla \Psi , \ \ j_1+j_2={k+1}, \ \ j_1,j_2\ge 1. \end{aligned}$$

We first treat the highest derivative term using the \(L^\infty \) smallness of small derivatives: Using (4.40) and (6.1)

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla \nabla \Psi |^2|\nabla ^{k_m}\nabla \Psi |^2\le ({{\mathscr {d}}}_0+b^2) I_{k_m}. \end{aligned}$$

We now assume \(j_1,j_2\le k_m-1\). If \(j_1\le \frac{4k_m}{9},\) then from (4.40), (7.4):

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla N_{j_1,j_2}|^2\lesssim ({{\mathscr {d}}}_0+b^2)\int \rho _{\mathrm{Tot}}^2\frac{|\nabla ^{j_2}\nabla \Psi |^2}{\langle Z\rangle ^{2(k+1-j_2)}}\le {{\mathscr {d}}}_0. \end{aligned}$$

The expression being symmetric in \(j_1,j_2\), we may assume \(j_1,j_2\ge m_0=\frac{4k_m}{9}+1\), \(j_1,j_2\le \frac{2k_m}{3}\), and using (4.40), (7.4):

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla N_{j_1,j_2}|^2\lesssim {{\mathscr {d}}}_0 \int _{Z\le Z^*}\frac{dZ}{\langle Z\rangle ^{\frac{k_m}{10}}}+b^4\int _{Z>Z^*}\frac{dZ}{\langle Z\rangle ^{\frac{k_m}{10}}}\le {{\mathscr {d}}}_0.\end{aligned}$$

Quantum pressure term. We estimate from Leibniz and (C.1):

$$\begin{aligned}&b^2\left| \Delta ^{K_m}\left( \frac{\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right) -\frac{\Delta ^{K_m+1} \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}+\frac{k\nabla \Delta ^{K_m} \rho _{\mathrm{Tot}}\cdot \nabla \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}^2}\right| \\&\quad \lesssim _k b^2\sum _{j_1+j_2=k,j_2\ge 2}\left| \nabla ^{j_1}\Delta \rho _{\mathrm{Tot}}\partial ^{j_2}\left( \frac{1}{\rho _{\mathrm{Tot}}}\right) \right| . \end{aligned}$$

We use the Faa di Bruno formula:

$$\begin{aligned} N_{j_1,j_2}=b^2\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}\frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{m_1+2m_2+\cdots +j_2m_{j_2}=j_2}\Pi _{i=1}^{j_2}(\nabla ^i\rho _{\mathrm{Tot}})^{m_i} \end{aligned}$$

and \(m_1+2m_2+\cdots +j_2m_{j_2}=j_2\). We decompose \({\rho _{\mathrm{Tot}}}=\rho _D+{\tilde{\rho }}\) in the sum and estimate the \(\rho _D\) contribution:

$$\begin{aligned}&b^4\int \rho _{\mathrm{Tot}}^2\left\{ \sum _{j_1+j_2=k,j_2\ge 2}\frac{|\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2}{\rho _{\mathrm{Tot}}^2\langle Z\rangle ^{2j_2}}+\frac{|\nabla ^{j_1}\Delta \rho _{\mathrm{Tot}}|^2}{\rho _{\mathrm{Tot}}^2\langle Z\rangle ^{2j_2+2}}\right\} \\&\quad \lesssim b^4\sum _{j_1+j_2=k,j_2\ge 2}\left[ \int \frac{\rho _{\mathrm{Tot}}^2dZ}{\langle Z\rangle ^{2j_2+2(j_1+3)}}+\int \frac{|\nabla ^{j_1+3}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2j_2}}\right] \\&\quad \lesssim b^4\left( 1+\sum _{j_1=2}^k\int \frac{|\nabla \nabla ^{j_1}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k-j_1)+2}}\right) \le {{\mathscr {d}}}_0+\delta J_{k_m}. \end{aligned}$$

In the general case, we replace \((\nabla ^i\rho _{\mathrm{Tot}})^{m_i}\) by \((\nabla ^i{\hat{\rho }})^{m_i}\) where \({\hat{\rho }}\) is either \(\rho _D\) or \({\tilde{\rho }}\). In both cases we will use the weaker estimates (4.40).

First, assume that \(m_i=0\) for \(i\ge \frac{4k_m}{9}+1\), then from (4.40):

$$\begin{aligned} |N_{j_1,j_2}|\lesssim & {} b^2|\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|\frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\sum _{m_1+2m_1+\cdots +j_2m_{j_2}=j_2}\Pi _{i=0}^{j_2}|(\nabla ^i{\hat{\rho }})^{m_i}| \\\lesssim & {} b^2 \frac{|\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|}{\rho _{\mathrm{Tot}}\langle Z\rangle ^{j_2}} \end{aligned}$$

and the conclusion follows as above. Otherwise, there are at most two value \(\frac{4k_m}{9}\le i_1\le i_2\le j_2\) with \(m_{i_1},m_{i_2}\ne 0\) and \(m_{i_1}+m_{i_2}\le 2\). Hence from (4.40):

$$\begin{aligned} \frac{1}{\rho _{\mathrm{Tot}}^{j_2+1}}\Pi _{i=0}^{j_2}|(\nabla ^i{\tilde{\rho }})^{m_i}|\lesssim & {} \frac{1}{\rho _D^{j_2+1}}|\nabla ^{i_1}{\tilde{\rho }}|^{m_{i_1}}|\nabla ^{i_2}{\hat{\rho }}|^{m_{i_2}}\Pi _{0\le i\le j_2, i\notin \{i_1,i_2\}}\left( \frac{\rho _D}{\langle Z\rangle ^{i}}\right) ^{m_i}\\\lesssim & {} \left( \frac{|\nabla ^{i_1}{\hat{\rho }}|}{\rho _D}\right) ^{m_{i_1}}\left( \frac{|\nabla ^{i_2}{\hat{\rho }}|}{\rho _D}\right) ^{m_{i_2}}\frac{1}{\rho _D\langle Z\rangle ^{j_2-(m_{i_1}i_1+m_{i_2}i_2)}}. \end{aligned}$$

Assume first \(i_2\ge \frac{2k_m}{3}+1\), then \(m_{i_1}=0\), \(m_{i_2}=1\) and \(j_1+3\le \frac{4k_m}{9}\) from which:

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \rho _{\mathrm{Tot}}^2 |\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\rho ^2_D}\frac{1}{\rho ^2_D\langle Z\rangle ^{2(j_2-i_2)}}\\\lesssim & {} b^4\int \frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(j_2-i_2)+2(j_1+3)}}\\\lesssim & {} b^4\int \frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(k-i_2)+6}}\le {\mathscr {d}}_0 \end{aligned}$$

There remains the case \(\frac{4k_m}{9}+1\le i_1\le i_2\le \frac{2k_m}{3}\) which imply \(j_1+3\le \frac{2k_m}{3}\), and we distinguish cases:

case \((m_{i_1},m_{i_2})=(0,1)\): if \(j_1+3\le \frac{4k_m}{9}\), we estimate

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \rho _D^2 |\nabla ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\rho ^2_D}\frac{1}{\rho ^2_D\langle Z\rangle ^{2(j_2-i_2)}}\\\lesssim & {} b^4\int \frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(j_2-i_2)+2(j_1+3)}}\\\lesssim & {} b^4\int \frac{|\nabla ^{i_2}{\hat{\rho }}|^2}{\langle Z\rangle ^{2(k-i_2)+6}}\le {{\mathscr {d}}}_0. \end{aligned}$$

Otherwise, \(\frac{4k_m}{9}+1\le j_1+3\le \frac{2k_m}{3}\). Hence \(\frac{4k_m}{9}+1\le j_1+3\le \frac{2k_m}{3}\), \(\frac{4k_m}{9}+1\le i_2\le \frac{2k_m}{3}\) and we estimate from (4.40), using \(k_m\) large:

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim b^4\int \frac{Z^{d-1} dZ}{\langle Z\rangle ^{2\left( \frac{k_m}{4}+\frac{k_m}{4}\right) }}\lesssim b^4\le {{\mathscr {d}}}_0. \end{aligned}$$

case \(m_{i_1}+m_{i_2}=2\): we obtain from (4.40) and \(j_1+3\le \frac{2k_m}{3}\)

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|N_{j_1,j_2}|^2\lesssim & {} b^4\int \rho _D^2 |\partial ^{j_1+1}\Delta \rho _{\mathrm{Tot}}|^2\left( \frac{1}{\langle Z\rangle ^{\frac{k_m}{4}}}\right) ^4\lesssim b^4\int \frac{dZ}{\langle Z\rangle ^{k_m}}\le {{\mathscr {d}}}_0. \end{aligned}$$

Step 6 \(\text {NL}({\tilde{\rho }})\) term. We need to estimate

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2\nabla \Delta ^{K_m}\text {NL}({\tilde{\rho }})\cdot \nabla \Psi _k \end{aligned}$$

which requires an integration by part in time for the highest order term. We expand using that, according to our assumptions, the nonlinearity is an integer:

$$\begin{aligned} \text {NL}({\tilde{\rho }})=(\rho _D+{\tilde{\rho }})^{p-1}-\rho _D^{p-1}-(p-1)\rho _D^{p-2}{\tilde{\rho }}=\sum _{q=2}^{p-1}c_{q}{\tilde{\rho }}^{q}\rho _D^{p-1-q} \end{aligned}$$

and hence by Leibniz:

$$\begin{aligned} \Delta ^{K_m}\text {NL}({\tilde{\rho }})= & {} \sum _{q=2}^{p-1}c_{q}{\tilde{\rho }}^{q-1}\left( \Delta ^{K_m}{\tilde{\rho }}\right) \rho _D^{p-1-q}\\&+ \sum _{q=2}^{p-1}\sum _{j_1+j_2=k}\sum _{\ell _1+\cdots +\ell _q=j_1, \ell _1\le \cdots \ell _q\le k-1}\nabla ^{\ell _1}{\tilde{\rho }}\cdots \nabla ^{\ell _q}{\tilde{\rho }}\nabla ^{j_2}(\rho _D^{p-1-q})\\ \end{aligned}$$

Let

$$\begin{aligned} N_{\ell _1,\ldots ,\ell _q,j_1,q}=\nabla ^{\ell _1}{\tilde{\rho }}\cdots \nabla ^{\ell _q}{\tilde{\rho }}\nabla ^{j_2}(\rho _D^{p-1-q}),\ \ \ell _1\le \cdots \le \ell _q. \end{aligned}$$

case \(\ell _q\le k_m-2\): we estimate

$$\begin{aligned} |\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim |\nabla ^{m_1}{\tilde{\rho }}\cdots \nabla ^{m_q}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}, \ \ \left| \begin{array}{ll} 0\le m_i\le k_m-1\\ m_1+\cdots m_q=j_1+1.\end{array}\right. \end{aligned}$$

We may reorder \(m_1\le \cdots \le m_q\). If \(m_q\le \frac{4k_m}{9}\), then

$$\begin{aligned} |\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim \frac{{\tilde{\rho }}^{q}}{\langle Z\rangle ^{j_1+1}}\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\lesssim \frac{{{\mathscr {d}}}_0}{\langle Z\rangle ^{\frac{k_m}{2}}} \end{aligned}$$

and hence the contribution of this term

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|^2\le {{\mathscr {d}}}_0. \end{aligned}$$

If \(\frac{4k_m}{9}\le m_q \le \frac{2k_m}{3}\), then similarly, combining (6.9), (6.1):

$$\begin{aligned} |\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim \frac{1}{\langle Z\rangle ^{j_2}}\frac{{{\mathscr {d}}}_0}{\langle Z\rangle ^{\frac{k_m}{4}}} \end{aligned}$$

and the conclusion follows. If \(m_q\ge \frac{2k_m}{3}\), then \(m_{q-1}\le \frac{4k_m}{9}\) from which:

$$\begin{aligned} |\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|\lesssim \frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\frac{\rho _D^{q-1}}{\langle Z\rangle ^{j_1+1-\ell _q}}|\nabla ^{\ell _q}{\tilde{\rho }}|\lesssim \frac{\rho _D^{p-2}|\nabla ^{\ell _q}{\tilde{\rho }}|}{\langle Z\rangle ^{k_m+1-m_q}} \end{aligned}$$

and hence the bound

$$\begin{aligned} \int \rho _{\mathrm{Tot}}^2|\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}|^2\lesssim & {} \int \rho _{\mathrm{Tot}}^{2(p-2)+2}\frac{|\nabla ^{m_q}{\tilde{\rho }}|^2}{\langle Z\rangle ^{2(k_m-m_q)+2}}\\\lesssim & {} \Vert {\tilde{\rho }},\Psi \Vert _{k_m-1,\sigma (k_m-1)}^2\le {{\mathscr {d}}}_0. \end{aligned}$$

case \(\ell _q= k_m-1\): we compute \(\nabla N_{\ell _1,\ldots ,\ell _q,j_1,q}\). If the derivative falls on \(\ell _j\), \(j\le q-1\), we are back to the previous case, and we are therefore left with estimating

$$\begin{aligned} |\partial ^{\ell _1}{\tilde{\rho }}\partial ^{\ell _{q-1}}{\tilde{\rho }}\partial ^{k_m}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}, \ \ \left| \begin{array}{ll} \ell _1+\cdots +\ell _{q-1}+k_m=j_1+1\\ j_1+j_2=k_m.\end{array}\right. \end{aligned}$$

If \(j_1=k_m-1\), then \(j_2=1\), \(\ell _1=\cdots =\ell _{q-1}=0\) and we estimate relying onto the smallness of \(\frac{{\tilde{\rho }}}{\rho _{\mathrm{Tot}}}\) from (7.4) (for \(Z\le Z^*\)) and using (4.40) together with the smallness of \(\langle Z\rangle ^{-1}\) (for \(Z\ge Z^*\)):

$$\begin{aligned} |\nabla ^{\ell _1}{\tilde{\rho }}\cdots \nabla ^{\ell _{q-1}}{\tilde{\rho }}\nabla ^{k_m}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\lesssim |{\tilde{\rho }}^{q-1} \nabla ^{k_m}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle }\lesssim {{\mathscr {d}}}_0 \rho _D^{p-{2}}|\nabla ^{k_m}{\tilde{\rho }}| \end{aligned}$$

and hence the corresponding contribution (\(p\ge 3\))

$$\begin{aligned} {{\mathscr {d}}}_0 \int \rho _D^{2(p-2)}|\partial ^{k_m}{\tilde{\rho }}|^2\le \delta J_{k_m}. \end{aligned}$$

Similarly, if \(j_1=k_m\) then \(\ell _1=\cdots =\ell _{q-2}=j_2=0\) and \(\ell _{q-1}=1\).

$$\begin{aligned} |\nabla ^{\ell _1}{\tilde{\rho }}\cdots \nabla ^{\ell _{q-1}}{\tilde{\rho }}\nabla ^{k_m}{\tilde{\rho }}|\frac{\rho _D^{p-1-q}}{\langle Z\rangle ^{j_2}}\lesssim & {} {\tilde{\rho }}^{q-2} |\nabla {\tilde{\rho }}| |\nabla ^{k_m}{\tilde{\rho }}| \rho _D^{p-1-q}\\\lesssim & {} {{\mathscr {d}}}_0 \rho _D^{p-2}|\nabla ^{k_m}{\tilde{\rho }}| \end{aligned}$$

Highest order term We are left with estimating the highest order term:

$$\begin{aligned} N_{\ell _1,\ldots ,\ell _q,j_1,q}={\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\Delta ^{K_m}{\tilde{\rho }}. \end{aligned}$$

We treat this term by integration by parts in time using (7.8):

$$\begin{aligned}&-\int \rho _{\mathrm{Tot}}^2\nabla \left[ {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\Delta ^{K_m}{\tilde{\rho }}\right] \cdot \nabla \Psi _k \nonumber \\&\quad =\int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\nabla \cdot (\rho _{\mathrm{Tot}}^2\nabla \Psi _k)\nonumber \\&\quad = -\int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\rho _{\mathrm{Tot}}\left[ \partial _\tau {\tilde{\rho }}_k-(H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k \right. \nonumber \\&\qquad +H_2\Lambda {\tilde{\rho }}_k+(\Delta ^{K_m}\rho _{\mathrm{Tot}})\Delta \Psi \nonumber \\&\qquad + \left. k\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k+2\nabla (\Delta ^{K_m}\rho _{\mathrm{Tot}})\cdot \nabla \Psi - F_1\right] \end{aligned}$$
(7.18)

and we treat all terms in (7.18). We will systematically use the smallness (4.27). The \(\partial _\tau {\tilde{\rho }}_k\) term is integrated by parts in time:

$$\begin{aligned}&-\int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}{\tilde{\rho }}_k\partial _\tau {\tilde{\rho }}_k=-\frac{1}{2}\frac{d}{d\tau }\left\{ \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right\} \\&\qquad + \frac{1}{2}\int {\tilde{\rho }}^2_k\partial _\tau \left( {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}\right) \\&\quad = -\frac{1}{2}\frac{d}{d\tau }\left\{ \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\right\} +O\left( \delta \int \rho _D^{p-1}{\tilde{\rho }}_k^2\right) \end{aligned}$$

and the boundary term in time is small

$$\begin{aligned} \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}{\tilde{\rho }}^2_k\lesssim \delta \int \rho _D^{p-1}\rho _k^2. \end{aligned}$$

We then estimate:

$$\begin{aligned}&\left| \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\rho _{\mathrm{Tot}}(H_1-k(H_2+\Lambda H_2)){\tilde{\rho }}_k\right| \\&\quad \lesssim k\delta \int \rho _{\mathrm{Tot}}^{p-1}{\tilde{\rho }}_k^2\lesssim {\delta }J_{k_m}.\end{aligned}$$

Using the extra decay in Z and \(\Vert \Delta \Psi \Vert _{L^\infty }\le \delta \ll 1\):

$$\begin{aligned}&\left| \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\rho _{\mathrm{Tot}}(\Delta ^{K_m}\rho _{\mathrm{Tot}})\Delta \Psi \right| \lesssim {{\mathscr {d}}}_0 \\&\quad \int \frac{dZ}{\langle Z\rangle ^{\frac{k_m}{2}}}+\int \rho _{\mathrm{Tot}}^{p-1}{\tilde{\rho }}^2_k|\Delta \Psi |\le {{\mathscr {d}}}_0+\delta J_{k_m}. \end{aligned}$$

Similarly, after an integration by parts:

$$\begin{aligned}&\left| -\int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\rho _{\mathrm{Tot}}\nabla (\Delta ^{K_m}\rho _{\mathrm{Tot}})\cdot \nabla \Psi \right| \lesssim {{\mathscr {d}}}_0 \\&\quad +\left| \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}\rho _{\mathrm{Tot}}\nabla ({\tilde{\rho }}_k^2)\cdot \nabla \Psi \right| \le {{\mathscr {d}}}_0+\delta J_{k_m}. \end{aligned}$$

Similarly, after an integration by parts using (4.40):

$$\begin{aligned}&\left| -\int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_k\rho _{\mathrm{Tot}}H_2\Lambda {\tilde{\rho }}_k\right| \lesssim {{\mathscr {d}}}_0 \\&\quad +\left( \left\| \frac{{\tilde{\rho }}}{\rho _{\mathrm{Tot}}}\right\| _{L^\infty }+\left\| \frac{Z |\nabla {\tilde{\rho }}|}{\rho _{\mathrm{Tot}}}\right\| _{L^\infty }\right) J_{k_m}\le {{\mathscr {d}}}_0+\delta J_{k_m} \end{aligned}$$

and similarly

$$\begin{aligned} \left| \int {\tilde{\rho }}^{q-1}\rho _D^{p-1-q}{\tilde{\rho }}_kk\nabla \rho _{\mathrm{Tot}}\cdot \nabla \Psi _k\right| \lesssim {{\mathscr {d}}}_0+\delta J_{k_m}. \end{aligned}$$

Step 7 Conclusion for \(k=k_m(d,r)\) large enough. We now sum the collection of above bounds and obtain the differential inequality with \(k=k_m\).

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ J_{k_m}(1+O(\delta ))\right\} \\&\quad \le -k\left[ 1+O\left( \frac{1}{k}\right) \right] \int ( H_2+\Lambda H_2)\\&\qquad \times \left[ b^2|\nabla {\tilde{\rho }}_k|^2+(p-1) \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_{k}^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right] \\&\qquad - k\int (p-1)\rho _D\partial _Z (\rho _D^{p-1}){\tilde{\rho }}_k\partial _Z \Psi _k+{{\mathscr {d}}}. \end{aligned}$$

We recall from (2.21), (2.22):

$$\begin{aligned} H_2+\Lambda H_2=\mu (1-w-\Lambda w)\ge c_{d,p}>0 \end{aligned}$$
(7.19)

and we now claim the pointwise coercivity of the coupled quadratic form: \(\exists c_{d,p}>0\) such that \(\forall Z\ge 0\),

$$\begin{aligned}&(H_2+\Lambda H_2)\left[ (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_{k}^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right] \nonumber \\&\qquad + (p-1)\rho _D\partial _Z (\rho _D^{p-1}){\tilde{\rho }}_k\partial _Z \Psi _k \nonumber \\&\quad \ge c_{d,p}\left[ (p-1) \rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_{k}^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right] \end{aligned}$$
(7.20)

which, after taking \(k>k^*(d,p)\) large enough, concludes the proof of (7.6).

Proof of (7.20). The coupling term is lower order for Z large:

$$\begin{aligned}&|(p-1)\rho _D\partial _Z (\rho _D^{p-1}){\tilde{\rho }}_k\partial _Z \Psi _k|\lesssim \frac{\rho _{\mathrm{Tot}}^{p-1}}{\langle Z\rangle }{\tilde{\rho }}_k\rho _{\mathrm{Tot}}\partial _Z\Psi _k \\&\quad \le \delta \left[ (p-1)\rho _D^{p-2}\rho _{\mathrm{Tot}}{\tilde{\rho }}_{k}^2+\rho _{\mathrm{Tot}}^2|\nabla \Psi _k|^2\right] \end{aligned}$$

for \(Z>Z(\delta )\) large enough. On a compact set using the smallness (4.27), (7.20) is implied by:

$$\begin{aligned}&(H_2+\Lambda H_2)\left[ (p-1)Q{\tilde{\rho }}_{k}^2+\rho _P^2|\nabla \Psi _k|^2\right] + (p-1)\rho _P\partial _Z Q{\tilde{\rho }}_k\partial _Z \Psi _k \nonumber \\&\quad \ge c_{d,p}\left[ (p-1)Q{\tilde{\rho }}_{k}^2+\rho _P^2|\nabla \Psi _k|^2\right] . \end{aligned}$$
(7.21)

We compute the discriminant:

$$\begin{aligned} \mathrm{Discr}= & {} (p-1)^2\rho _P^2(\partial _ZQ)^2-4\mu ^2(p-1)\rho _P^2Q(H_2+\Lambda H_2)^2\\= & {} (p-1)\rho _P^2 Q\left[ (p-1)\frac{(\partial _ZQ)^2}{Q}-4\mu ^2(1-w-\Lambda w)^2\right] . \end{aligned}$$

We compute from (2.10):

$$\begin{aligned} (p-1)\frac{(\partial _ZQ)^2}{Q}= & {} (p-1)\left( 2\partial _Z\sqrt{Q}\right) ^2 \\= & {} (p-1)\left( \frac{1-{\mathscr {e}}}{2}\sqrt{\ell }\partial _Z({\sigma } Z)\right) ^2=(1-{\mathscr {e}})^2(\partial _Z(Z{\sigma }))^2\\= & {} \frac{4}{r^2}(\partial _Z(Z{\sigma }))^2=4\mu ^2{(\sigma +\Lambda \sigma )^2} \end{aligned}$$

and hence from (2.21), (2.22) the lower bound:

$$\begin{aligned} -\mathrm{Discr}= & {} 4\mu ^2(p-1)\rho _P^2 Q\left[ (1-w-\Lambda w)^2-{(\sigma +\Lambda \sigma )^2}\right] \\\ge & {} c_{d,r}(p-1)\rho _P^2Q, \ \ c_{d,r}>0, \end{aligned}$$

which together with (7.19) concludes the proof of (7.20). \(\square \)

8 Control of low Sobolev norms and proof of Theorem 1.1

Our aim in this section is to control weighted low Sobolev norms in the interior \(r\le 1\) (\(Z\le Z^*\)). On our way we will conclude the proof of the bootstrap Proposition 4.4. Theorem 1.1 will then follow from a classical topological argument.

8.1 Exponential decay slightly beyond the light cone

We use the exponential decay estimate (3.5) for a linear problem to prove exponential decay for the nonlinear evolution in the region slightly past the light cone \(Z=Z_2\). We recall the notations of Sect. 3, in particular \(Z_a\) of Lemma 3.7.

Lemma 8.1

(Exponential decay slightly past the light cone) Let

$$\begin{aligned} \tilde{Z_a}=\frac{Z_2+Z_a}{2}. \end{aligned}$$

Then, there holds the following bound:

$$\begin{aligned} \Vert \nabla \Phi \Vert _{H^{2k_0}(Z\le \tilde{Z_a})}+\Vert \rho \Vert _{H^{2k_0}(Z\le \tilde{Z_a})}\lesssim e^{-\frac{\delta _g}{2}\tau }. \end{aligned}$$
(8.1)

Proof

The proof relies on the spectral theory beyond the light cone \(Z=Z_2\) and an elementary finite speed propagation like argument in renormalized variables, related to [48].

Step 1 Semigroup decay in X variables. Recall the definition (4.12) of \(X=(\Phi ,T)\)

$$\begin{aligned} \left| \begin{array}{ll} \Phi =\rho _P\Psi \\ T=\partial _\tau \Phi +aH_2\Lambda \Phi \\ \quad =-(p-1)Q\rho -H_2\Lambda \Phi +(H_1-{\mathscr {e}})\Phi +G_\Phi +aH_2\Lambda \Phi \end{array}\right. \end{aligned}$$
(8.2)

with \(G_\Phi \) given by (3.11), the scalar product (3.44) and the definitions (4.14), (4.15):

$$\begin{aligned} \left| \begin{array}{l} \Lambda _0=\{\lambda \in {\mathbb {C}}, \ \ \mathfrak {R}(\lambda )\ge 0\} \cap \{\lambda \ \ \text{ is } \text{ an } \text{ eigenvalue } \text{ of }\ \ {\mathcal {M}}\}=(\lambda _i)_{1\le i\le N},\\ V=\cup _{1\le i\le N} \text{ ker } ({\mathcal {M}}-\lambda _i I)^{k_{\lambda _i}} \end{array}\right. \end{aligned}$$

the projection \({{\mathbb {P}}}\) associated with V, the decay estimate (3.5) on the range of \((I-{{\mathbb {P}}})\) and the results of Lemma 3.5. Relative to the X variables our equations take the form

$$\begin{aligned} \partial _\tau X={\mathcal {M}} X + G, \end{aligned}$$

which are considered on the time interval \(\tau \ge \tau _0\gg 1\) and the space interval \(Z\in [0,Z_a]\) (no boundary conditions at \(Z_a\)). We consider evolution in the Hilbert space \({{\mathbb {H}}_{2k_0}}\) with initial data such that

$$\begin{aligned} \Vert (I-{{\mathbb {P}}}) X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{\delta _g}{2}\tau _0},\qquad \Vert {{\mathbb {P}}} X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{3\delta _g}{5}\tau _0}. \end{aligned}$$
(8.3)

According to the bootstrap assumption (4.45)

$$\begin{aligned} \Vert {{\mathbb {P}}}X(\tau )\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{\delta _g}{2}\tau }, \qquad \forall \tau \in [\tau _0,\tau ^*] \end{aligned}$$
(8.4)

Lemma 3.5 shows that as long as

$$\begin{aligned} \Vert G\Vert _{{\mathbb {H}}_{2k_0}}\le e^{-\frac{2\delta _g}{3}\tau }, \qquad \tau \ge \tau _0, \end{aligned}$$
(8.5)

there exists \(\Gamma \), which can be made as large as we want with a choice of \(\tau _0\), such that

$$\begin{aligned} \Vert {{\mathbb {P}}}X(\tau )\Vert _{{\mathbb {H}}_{2k_0}}\lesssim e^{-\frac{\delta _g}{2}\tau },\qquad \tau _0\le \tau \le \tau _0+\Gamma . \end{aligned}$$
(8.6)

This will allow us to show eventually that if we can verify (8.5), the bootstrap time \(\tau ^*\ge \tau _0+\Gamma \).

Moreover, as long as (8.5) holds, the decay estimate (3.5) implies that

$$\begin{aligned} \Vert (I-{{\mathbb {P}}})X(\tau )\Vert _{{\mathbb {H}}_{2k_0}}\lesssim & {} e^{-\frac{\delta _g}{2}(\tau -\tau _0)}\Vert X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}\nonumber \\&+\int _{\tau _0}^{\tau }e^{-\frac{\delta _g}{2}(\tau -\sigma )}\Vert G(\sigma )\Vert _{{\mathbb {H}}_{2k_0}}d\sigma \nonumber \\\lesssim & {} e^{\frac{-\delta _g}{2}\tau }\left[ e^{\frac{\delta _g}{2}\tau _0}\Vert X(\tau _0)\Vert _{{\mathbb {H}}_{2k_0}}+\int _{\tau _0}^{+\infty }e^{-\frac{\delta _g}{6}\tau }d\tau \right] \nonumber \\\le & {} e^{-\frac{\delta _g}{2}\tau }. \end{aligned}$$
(8.7)

As a result,

$$\begin{aligned} \Vert X(\tau )\Vert _{{\mathbb {H}}_{2k_0}}\lesssim e^{-\frac{\delta _g}{2}\tau },\qquad \tau _0\le \tau \le \tau ^*. \end{aligned}$$
(8.8)

Below we will verify (8.5) \(\forall \tau \in [\tau _0,\tau ^*]\) under the assumption (8.7), closing both. Once again, this will allow us to show eventually that the length of the bootstrap interval \(\tau ^*-\tau _0\ge \Gamma \) is sufficiently large.

Recall from (3.13), (3.14), (3.44):

$$\begin{aligned} \Vert G\Vert ^2_{{\mathbb {H}}_{2k_0}}\lesssim \int _{Z\le Z_a} |\nabla \Delta ^{k_0}G_T|^2gZ^{d-1}dZ+\int _{Z\le Z_a} G_T^2Z^{d-1}dZ \end{aligned}$$
(8.9)

with

$$\begin{aligned} \left| \begin{array}{lll} G_T=\partial _\tau G_\Phi -\left( H_1+H_2\frac{\Lambda Q}{Q}\right) G_\Phi +H_2\Lambda G_\Phi -(p-1)QG_\rho ,\\ G_\rho =-\rho \Delta \Psi -2\nabla \rho \cdot \nabla \Psi ,\\ G_\Phi =-\rho _P(|\nabla \Psi |^2+\text {NL}(\rho ))+\frac{b^2\rho _P}{\rho _{\mathrm{Tot}}}\Delta \rho _{\mathrm{Tot}}. \end{array}\right. \end{aligned}$$

Step 2 Semigroup decay for \((\rho ,\Psi )\). We now translate the X bound to the bounds for \(\rho \) and \(\Psi \) and then verify (8.5). We recall (8.2) and obtain for any \({{\hat{Z}}}>Z_2\)

$$\begin{aligned}&\Vert T\Vert _{H^{2k_0}(Z\le {\hat{Z}})}+\Vert \Phi \Vert _{H^{2k_0+1}(Z\le {\hat{Z}})} \\&\quad \lesssim \Vert \rho \Vert _{H^{2k_0}(Z\le {\hat{Z}})}+\Vert \Psi \Vert _{H^{2k_0+1}(Z\le {\hat{Z}})} +\Vert G_\Phi \Vert _{H^{2k_0}(Z\le {\hat{Z}})}\\&\quad \lesssim \Vert T\Vert _{H^{2k_0}(Z\le {\hat{Z}})}+\Vert \Phi \Vert _{H^{2k_0+1}(Z\le {\hat{Z}})}+\Vert G_\Phi \Vert _{H^{2k_0}(Z\le {\hat{Z}})} \end{aligned}$$

and claim:

$$\begin{aligned} \Vert G_{\Phi }\Vert _{H^{2k_0}(Z\le {\hat{Z}})}\lesssim \Vert \nabla \Psi \Vert _{H^{2k_0}(Z\le {\hat{Z}})}^2+\Vert \rho \Vert _{H^{2k_0}(Z\le {\hat{Z}})}^2+e^{-{\delta _g}\tau }. \end{aligned}$$
(8.10)

Indeed, since \(H^{2k_0}(Z\le {\hat{Z}})\) is an algebra for \(k_0\) large enough:

$$\begin{aligned} \Vert \rho _P(|\nabla \Psi |^2+\text {NL}(\rho ))\Vert _{H^{2k_0}(Z\le {\hat{Z}})}\lesssim \Vert \nabla \Psi \Vert _{H^{2k_0}(Z\le {\hat{Z}})}^2+\Vert \rho \Vert _{H^{2k_0}(Z\le {\hat{Z}})}^2. \end{aligned}$$

The remaining quantum pressure term is treated using the pointwise bound (4.40) for small Sobolev norms and the smallness of b which imply:

$$\begin{aligned} \left\| \frac{b^2\rho _P\Delta \rho _{\mathrm{Tot}}}{\rho _{\mathrm{Tot}}}\right\| _{H^{2k_0}(Z\le {\hat{Z}})}\lesssim C_Kb^2\le e^{-\delta _g\tau } \end{aligned}$$

provided \(\delta _g>0\) has been chosen small enough, and (8.10) is proved. Choosing \({{\hat{Z}}}>Z_2\), this implies from (8.2) and the initial bound (4.19):

$$\begin{aligned} \Vert X(\tau _0)\Vert _{{\mathbb {H}}^{2k_0}}\lesssim & {} \Vert \Psi (\tau _0)\Vert _{H^{2k_0+1}(Z\le {\hat{Z}})}+\Vert \rho (\tau _0)\Vert _{H^{2k_0}(Z\le {\hat{Z}})}+e^{-\delta _g\tau _0} \nonumber \\\lesssim & {} e^{-\frac{\delta _g\tau _0}{2}}. \end{aligned}$$
(8.11)

This verifies (8.3). On the other hand, choosing \({{\hat{Z}}}={{\tilde{Z}}}_a\) with

$$\begin{aligned} \tilde{Z_a}=\frac{Z_2+Z_a}{2}, \end{aligned}$$

we also obtain from (8.8)

$$\begin{aligned} \Vert \Psi (\tau )\Vert _{H^{2k_0+1}(Z\le {\tilde{Z}}_a)}+\Vert \rho (\tau )\Vert _{H^{2k_0}(Z\le {\tilde{Z}}_a)} \lesssim \Vert X(\tau )\Vert _{{\mathbb {H}}^{2k_0}}+ e^{-\delta _g\tau }\lesssim e^{-\frac{\delta _g\tau }{2}}.\nonumber \\ \end{aligned}$$
(8.12)

The estimate (8.1) follows.

Step 3 Estimate for G. Proof of (8.5). We recall (8.9). On a fixed compact domain \(Z\le Z_0\) with \(Z_0>Z_2\), we can interpolate the bootstrap bound (4.39) with the global large Sobolev bound (4.38) and obtain for \(k_m\) large enough and \(b_0<b_0(k_m)\) small enough:

$$\begin{aligned} \Vert \rho \Vert _{H^{2k_0+10}(Z\le Z_0)}+\Vert \Psi \Vert _{H^{2k_0+10}(Z\le Z_0)}\le C_Ke^{-\left[ \frac{3}{8}-\frac{1}{100}\right] \delta _g\tau }\le e^{-\left[ \frac{3}{8}-\frac{1}{50}\right] \delta _g\tau }\nonumber \\ \end{aligned}$$
(8.13)

and since \(H^{2k_0}\) is an algebra and all terms are either quadratic or with a b term, (8.13) implies

$$\begin{aligned}&\Vert G_T\Vert _{H^{2k_0+5}(Z\le Z_0)}+\Vert G_\rho \Vert _{H^{2k_0+5}(Z\le Z_0)}+\Vert G_\Phi \Vert _{H^{2k_0+5}(Z\le Z_0)} \nonumber \\&\quad \le e^{-\left( \frac{3}{4}-\frac{1}{20}\right) \delta _g\tau }\le e^{-\frac{2\delta _g}{3}\tau }, \end{aligned}$$
(8.14)

which in particular using (8.9) implies (8.5). \(\square \)

8.2 Weighted decay for \(m\le 2k_0\) derivatives

We recall the notation (3.8). We now transform the exponential decay (8.1) from just past the light cone into weighted decay estimate. It is essential for this argument that the decay (8.1) has been shown in the region strictly including the light cone \(Z=Z_2\). The estimates in the lemma below close the remaining bootstrap bound (4.39).

Lemma 8.2

(Weighted Sobolev bound for \(m\le 2k_0\)) Let \(m\le 2k_0\) and \(\nu _0=\frac{\delta _g}{2\mu }-\frac{2(r-1)}{p-1}\), recall

$$\begin{aligned} \chi _{\nu _0,m}=\frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)}}\zeta \left( \frac{Z}{Z^*}\right) , \ \ \zeta (Z)=\left| \begin{array}{ll}1\ \ \text{ for }\ \ Z\le 2\\ 0\ \ \text{ for }\ \ Z\ge 3, \end{array}\right. \end{aligned}$$

then

$$\begin{aligned} \sum _{m=0}^{2k_0}\sum _{i=1}^d\int (p-1)Q(\partial ^m_i\rho )^2\chi _{\nu _0,m}+|\nabla \partial ^m_i\Phi |^2\chi _{\nu _0,m}\le C e^{-\frac{4\delta _g}{5}\tau }. \end{aligned}$$
(8.15)

Proof of Lemma 8.2

The proof relies on a sharp energy estimate with time dependent localization of \((\rho ,\Phi )\). This is a renormalized version of the finite speed of propagation.

Step 1 \({\dot{H}}^{m}\) localized energy identity. Pick a smooth well localized radially symmetric function \(\chi (\tau ,Z)\) and a coordinate \(1\le i\le d\) and note for m integer

$$\begin{aligned} \rho _m=\partial ^m_i\rho , \ \ \Phi _m=\partial ^m_i\Phi , \end{aligned}$$

where we omit the i dependence to simplify notations. We recall the Emden transform formulas (2.24):

$$\begin{aligned} \left| \begin{array}{ll} H_2=\mu (1-w),\\ H_1=\frac{\mu \ell }{2}(1-w)\left[ 1+\frac{\Lambda \sigma }{\sigma }\right] ,\\ H_3=\frac{\Delta \rho _P}{\rho _P}, \end{array}\right. \end{aligned}$$
(8.16)

which yield the bounds using (2.19), (2.20):

$$\begin{aligned} \left| \begin{array}{llll} H_2=\mu +O\left( \frac{1}{\langle Z\rangle ^r}\right) , \ \ H_1=-\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^r}\right) ,\\ |\langle Z\rangle ^j\partial _Z^j H_1|+||\langle Z\rangle ^j\partial _Z^jH_2|\lesssim \frac{1}{\langle Z\rangle ^{r}}, \ \ j\ge 1,\\ |\langle Z\rangle ^j\partial _Z^j H_3|\lesssim \frac{1}{\langle Z\rangle ^2},\\ \frac{1}{\langle Z\rangle ^{2(r-1)}}\left[ 1+O\left( \frac{1}{\langle Z\rangle ^{r}}\right) \right] \lesssim _j |\langle Z\rangle ^j\partial _Z^jQ|\lesssim _j \frac{1}{\langle Z\rangle ^{2(r-1)}} \end{array}\right. \end{aligned}$$
(8.17)

and the commutator bounds:

$$\begin{aligned} \left| \begin{array}{lllll} |[\partial _i^m,H_1]\rho |\lesssim \sum _{j=0}^{m-1}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{r+m-j}},\\ |\nabla \left( [\partial _i^m,H_1]\rho \right) |\lesssim \sum _{j=0}^{m}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{m-j+r+1}},\\ |[\partial _i^m,Q]\rho |\lesssim Q\sum _{j=0}^{m-1}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{m-j}},\\ |[\partial _i^m,H_2]\Lambda \rho |\lesssim \sum _{j=1}^{m}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{r+m-j}},\\ |\nabla \left( [\partial _i^m,H_2]\Lambda \Phi \right) |\lesssim \sum _{j=1}^{m+1}\frac{|\partial ^j_Z\Phi |}{\langle Z\rangle ^{r+1+m-j}}. \end{array}\right. \end{aligned}$$
(8.18)

Commuting (3.9) with \(\partial _i^m\):

$$\begin{aligned} \left| \begin{array}{ll} \partial _\tau \rho _m=H_1\rho _m-H_2(m+\Lambda )\rho _m-\Delta \Phi _m+\partial _i^mG_\rho +E_{m,\rho },\\ \partial _\tau \Phi _m=-(p-1)Q\rho _m-H_2(m+\Lambda ) \Phi _m+(H_1-{\mathscr {e}})\Phi _m+\partial ^m_iG_\Phi +E_{m,\Phi } \end{array}\right. \end{aligned}$$

with the bounds

$$\begin{aligned} \left| \begin{array}{ll} |E_{m,\rho }|\lesssim \sum _{j=0}^{m}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{r-1+m-j}}+\sum _{j=0}^{m}\frac{|\partial _Z^j\Phi |}{\langle Z\rangle ^{m-j+2}},\\ |\nabla E_{m,\Phi }|\lesssim Q\sum _{j=0}^{m}\frac{|\partial _Z^j\rho |}{\langle Z\rangle ^{m+1-j}}+ \sum _{j=0}^{m+1}\frac{|\partial ^j_Z\Phi |}{\langle Z\rangle ^{r+m-j}}. \end{array}\right. \end{aligned}$$

Let \(\chi \) be an arbitrary smooth function. We derive the corresponding energy identity:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int (p-1)Q\rho _m^2\chi +|\nabla \Phi _m|^2\chi \right\} \\&\quad =\frac{1}{2}\int \partial _\tau \chi \left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \\&\qquad + \int (p-1)Q\rho _m\chi \left[ H_1\rho _m-H_2(m+\Lambda )\rho _m \right. \\&\qquad \left. -\Delta \Phi _m+\partial _i^mG_\rho +E_{m,\rho }\right] \\&\qquad + \int \chi \nabla \Phi _m\cdot \nabla \left[ -(p-1)Q\rho _m-H_2(m+\Lambda \Phi _m)\right. \\&\qquad \left. +(H_1-\mu (r-2))\Phi _m+\partial ^i_mG_\Phi +E_{m,\Phi }\right] \\&\quad = \frac{1}{2}\int \partial _\tau \chi \left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \\&\qquad + \int (p-1)Q\rho _m\chi \left[ H_1\rho _m-H_2(m+\Lambda )\rho _m\right. \\&\qquad \left. +\partial _i^mG_\rho +E_{m,\rho }\right] +\int (p-1)Q\rho _m\nabla \chi \cdot \nabla \Phi _m\\&\qquad + \int \chi \nabla \Phi _m\cdot \nabla \left[ -H_2(m+\Lambda ) \Phi _m\right. \\&\qquad \left. +(H_1-\mu (r-2))\Phi _m+\partial ^m_iG_\Phi +E_{m,\Phi }\right] .\\ \end{aligned}$$

In what follows we will use \(\omega >0\) as a small universal constant to denote the power of tails of the error terms. In most cases, the power is in fact \(r>2\) which we do not need.

\(\rho _m\) terms. From the asymptotic behavior of Q (2.20) and (8.17):

$$\begin{aligned}&-\int (p-1)Q\rho _m\chi H_2\Lambda \rho _m\\&\quad =\frac{p-1}{2}\int \rho _m^2 \chi QH_2\left[ d+\frac{\Lambda Q}{Q}+\frac{\Lambda H_2}{H_2}+\frac{\Lambda \chi }{\chi }\right] \\&\quad = \int \rho _m^2 (p-1)\chi Q\mu \left[ \frac{d}{2}-(r-1)+O\left( \frac{1}{\langle Z\rangle ^\omega }\right) \right] \\&\qquad +\frac{1}{2}\int (p-1)QH_2\Lambda \chi \rho _m^2 \end{aligned}$$

\(\Phi _m\) terms. We first estimate recalling (8.17):

$$\begin{aligned}&\int \chi \nabla \Phi _m\cdot \nabla \left[ (-mH_2+H_1-\mu (r-2))\Phi _m\right] \\&\quad =\int (-mH_2+H_1-\mu (r-2))\chi |\nabla \Phi _m|^2\\&\qquad +O\left( \int \frac{\chi }{\langle Z\rangle ^{r}}|\nabla \Phi _m||\Phi _m|\right) \\&\quad = -\left[ \mu (m+r-2)+\frac{2\mu (r-1)}{p-1}\right] \int \chi |\nabla \Phi _m|^2\\&\qquad +O\left( \int \frac{\chi }{\langle Z\rangle ^\omega }\left[ |\nabla \Phi _m|^2+\frac{\Phi _m^2}{\langle Z\rangle ^2}\right] \right) . \end{aligned}$$

From Pohozhaev identity (5.27) with \(F=\chi H_2(Z_1,\ldots ,Z_d)\):

$$\begin{aligned}&-\int \chi \nabla \Phi _m\cdot \nabla (H_2\Lambda \Phi _m)=\int H_2\Lambda \Phi _m[ \chi \Delta \Phi _m+\nabla \chi \cdot \nabla \Phi _m]\\&\quad = -\sum _{i,j=1}^d \int \partial _iF_j\partial _i\Phi _m\partial _j\Phi _m+\frac{1}{2}\int |\nabla \Phi _m|^2\nabla \cdot F\\&\qquad +\int H_2\Lambda \Phi _m\nabla \chi \cdot \nabla \Phi _m\\&\quad = \sum _{i,j=1}^d\partial _i\Phi _m\partial _j\Phi _m\left[ -\partial _i(\chi H_2 Z_j)+H_2Z_j\partial _i\chi \right] \\&\qquad +\frac{1}{2}\int |\nabla \Phi _m|^2\chi H_2\left[ d+\frac{\Lambda \chi }{\chi }+\frac{\Lambda H_2}{H_2}\right] \\&\quad =\frac{\mu (d-2)}{2}\int \chi |\nabla \Phi _m|^2+\frac{1}{2} \int H_2\Lambda \chi |\nabla \Phi _m|^2 +O\left( \int \frac{\chi }{\langle Z\rangle ^\omega }|\nabla \Phi _m|^2\right) . \end{aligned}$$

The collection of above bounds yields for some universal constant \(\omega >0\) the weighted energy identity:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int (p-1)Q\rho _m^2\chi +|\nabla \Phi _m|^2\chi \right\} \nonumber \\&\quad =-\int \chi \left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \nonumber \\&\qquad \times \left[ \mu \left( m-\frac{d}{2}+r-1\right) +\frac{2\mu (r-1)}{p-1}+O\left( \frac{1}{\langle Z\rangle ^\omega }\right) \right] \nonumber \\&\qquad + \frac{1}{2}\int (p-1)Q\rho _m^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] \nonumber \\&\qquad +\frac{1}{2}\int |\nabla \Phi _m|^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] +\int (p-1)Q\rho _m\nabla \chi \cdot \nabla \Phi _m\nonumber \\&\qquad + O\left( \int \chi \left[ \sum _{j=0}^{m+1}\frac{|\partial _Z^j\Phi |^2}{\langle Z\rangle ^{2(m+1-j)+\omega }}+\sum _{j=0}^{m}\frac{Q|\partial _Z^j\rho |^2}{\langle Z\rangle ^{2(m-j)+\omega }}\right] \right) \nonumber \\&\qquad + O\left( \int \chi |\nabla \Phi _m||\nabla \partial ^mG_\Phi |+\int \chi Q|\rho _m||\partial ^mG_\rho |\right) . \end{aligned}$$
(8.19)

Step 2 Nonlinear and source terms. We claim the bound for \(\chi =\chi _{\nu _0,m}\):

$$\begin{aligned}&\sum _{m=0}^{2k_0} \sum _{i=1}^d\int \chi _{\nu _0,m}|\nabla \partial ^mG_\Phi |^2+\int (p-1)Q\chi _{\nu _0,m}|\partial ^mG_\rho |^2 \nonumber \\&\quad \lesssim \left( \sum _{m=0}^{2k_0}\sum _{i=1}^d\int Q\rho _m^2\chi _{\nu _0+1,m}+|\nabla \Phi _m|^2\chi _{\nu _0+1,m}\right) +b^2. \end{aligned}$$
(8.20)

\(G_\rho \) term. Recall ()

$$\begin{aligned} G_\rho =-\rho \Delta \Psi -2\nabla \rho \cdot \nabla \Psi , \end{aligned}$$

then by Leibniz:

$$\begin{aligned} |\partial ^mG_\rho |^2\lesssim \sum _{j_1+j_2=m+2, j_2\ge 1}|\partial ^{j_1}\rho |^2|\partial ^{j_2}\Psi |^2. \end{aligned}$$

We recall the pointwise bounds (4.40) for \(Z\le 3Z^*\),

$$\begin{aligned} |\partial ^{j_1}\rho |\le \frac{C_K}{\langle Z\rangle ^{j_1+\frac{2(r-1)}{p-1}}}, \ \ |\partial ^{j_2}\Psi |\le & {} \frac{C_K}{\langle Z\rangle ^{j_2+r-2}}. \end{aligned}$$

This yields, recalling (8.33), for \(j_1\le 2k_0\):

$$\begin{aligned}&\int \chi _{\nu _0,m}Q|\partial ^{j_1}\rho |^2|\partial ^{j_2}\Psi |^2\lesssim \int Q\zeta \left( \frac{Z}{Z^*}\right) \frac{|\partial ^{j_1}\rho |^2}{Z^{2(j_2-m)+d-2(r-1)+2(r-2)+2\nu _0}}\\&\quad \lesssim \int \zeta \left( \frac{Z}{Z^*}\right) Q\frac{|\partial ^{j_1}\rho |^2}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-j_1)+2}}\lesssim \sum _{j=0}^{j_1}\int \chi _{\nu _0+1,j_1}Q|\partial _Z^{j}\rho |^2\\&\quad \lesssim \sum _{m=0}^{2k_0}\sum _{i=1}^d\int Q\rho _m^2\chi _{\nu _0+1,m}+|\nabla \Phi _m|^2\chi _{\nu _0+1,m}. \end{aligned}$$

For \(j_1=m+1\), \(j_2=1\), we use the other variable:

$$\begin{aligned}&\int \chi _{\nu _0,m}Q|\partial ^{j_1}\rho |^2|\partial ^{j_2}\Psi |^2\lesssim \int Q\zeta \left( \frac{Z}{Z^*}\right) \frac{|\partial ^{j_2}\Psi |^2}{Z^{2(j_1-m)+d-2(r-1)+\frac{4(r-1)}{p-1}+2\nu _0}}\\&\quad \lesssim \int \zeta \left( \frac{Z}{Z^*}\right) \frac{\rho _P^2|\partial ^{j_2}\Psi |^2}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-j_2)+2}}\\&\quad \lesssim \sum _{j=0}^{j_2}\int \zeta \left( \frac{Z}{Z^*}\right) \frac{|\partial _Z^{j}\Phi |^2}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-j)+2}}\\&\quad \lesssim \sum _{j=0}^{j_2}\int \chi _{\nu _0+1,j}|\partial _Z^{j}\Phi |^2\lesssim \sum _{m=0}^{2k_0}\sum _{i=1}^d\int Q\rho _m^2\chi _{\nu _0+1,m}+|\nabla \Phi _m|^2\chi _{\nu _0+1,m} \end{aligned}$$

and (8.20) follows for \(G_\rho \) by summation on \(0\le m\le 2k_0\).

\(G_\Phi \) term. Recall (3.11)

$$\begin{aligned} G_\Phi =-\rho _P(|\nabla \Psi |^2+\text {NL}(\rho ))+\frac{b^2\rho _P}{\rho _{\mathrm{Tot}}}\Delta \rho _{\mathrm{Tot}}. \end{aligned}$$

We estimate using the pointwise bounds (4.40) for \(j_3\le 2k_0\):

$$\begin{aligned}&|\nabla \partial ^m(\rho _P|\nabla \Psi |^2)|\lesssim \sum _{j_1+j_2+j_3=m+1,j_2\le j_3}\frac{\rho _P}{\langle Z\rangle ^{j_1}}|\partial ^{j_2+1}\Psi \partial ^{j_3+1}\Psi |\\&\quad \lesssim \sum _{j_1+j_2+j_3=m+1,j_2\le j_3}\frac{1}{\langle Z\rangle ^{\frac{2(r-1)}{p-1}+j_1+r-2+j_2+1}}|\partial ^{j_3+1}\Psi |\\&\quad \lesssim \sum _{j_3=0}^{2k_0}\frac{|\partial ^{j_3+1}\Phi |}{\langle Z\rangle ^{r+m-j_3}} \end{aligned}$$

and since \(r>1\):

$$\begin{aligned} \sum _{j_3=0}^{2k_0}\int \chi _{\nu _0,m}\frac{|\partial ^{j_3+1}\Phi |^2}{\langle Z\rangle ^{2(r+m-j_3)}}\lesssim \sum _{j_3=0}^{2k_0}\int \chi _{\nu _0+1,j_3}|\nabla \Phi _{j_3}|^2. \end{aligned}$$

For \(j_3=2k_0+1\), we use the other variable and the conclusion follows similarly.

The quantum pressure term is estimated using the pointwise bounds (4.40):

$$\begin{aligned}&\int \chi _{\nu _0,m}\left| \nabla \partial ^m\left( \frac{b^2\rho _P}{\rho _{\mathrm{Tot}}}\Delta \rho _{\mathrm{Tot}}\right) \right| ^2\lesssim C_Kb^4\int _{Z\le 3Z^*}\frac{\chi _{\nu _0,m}}{\langle Z\rangle ^{\frac{4(r-1)}{p-1}+2(m+3)}}\\&\quad \lesssim C_K b^4 \int _{Z\le 3Z^*}\frac{Z^{d-1}dZ}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0+\frac{2(r-1)}{p-1}-m)+2(m+3)}}\le b^2. \end{aligned}$$

Step 2 Initialization and lower bound on the bootstrap time \(\tau ^*\).

Fix a large enough \(Z_0\) and pick a small enough universal constant \(\omega _0\) such that

$$\begin{aligned} \forall Z\ge 0, \ \ -\omega _0+H_2\ge \frac{\omega _0}{2}>0 \end{aligned}$$
(8.21)

and let \(\Gamma =\Gamma (Z_0)\) such that

$$\begin{aligned} \frac{Z_0}{2\hat{Z_a}}e^{-\omega _0\Gamma }=1. \end{aligned}$$
(8.22)

We claim that provided \({\tau _0}\) has been chosen sufficiently large, the bootstrap time \(\tau ^*\) of Proposition 4.4 satisfies the lower bound

$$\begin{aligned} \tau ^*\ge \tau _0+\Gamma . \end{aligned}$$
(8.23)

Indeed, in view of Sects. 567 there remains to control the bound (4.39) on \([\tau _0,\tau _0+\Gamma ]\). By (8.6) (8.7), the desired bounds already hold for \(Z\le {{\tilde{Z}}}_a\) on \([\tau _0,\tau _0+\Gamma ]\).

We now run the energy estimate (8.19) with \(\chi =\chi _{\nu _0,m}\) and obtain from (8.19), (8.20) the rough bound on \([\tau _0,\tau ^*]\):

$$\begin{aligned}&\frac{d}{d\tau }\left\{ \int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right\} \\&\quad \le C\int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}+b^2. \end{aligned}$$

which yields using (4.19):

$$\begin{aligned}&\int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\\&\quad \le e^{C(\tau -\tau _0)} \int (p-1)Q(\rho _m(0))^2\chi _{\nu _0,m}+|\nabla \Phi _m(0)|^2\chi _{\nu _0,m}\\&\qquad + e^{C\tau }\int _{\tau _0}^{\tau }e^{-(C+2\delta _g)\sigma }d\sigma \\&\quad \le e^{C\Gamma }\left[ C_0e^{-\delta _g\tau _0}+e^{-2\delta _g\tau _0}\right] \le 2e^{C\Gamma }C_0e^{-\delta _g\tau _0} \end{aligned}$$

and hence

$$\begin{aligned}&e^{\frac{4\delta _g}{5}\tau }\left[ \int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right] \\&\quad \le e^{\frac{4\delta _g}{5}\tau _0}e^{\frac{4\delta _g}{5}\Gamma }\left[ \int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m} \right. \\&\qquad \left. \times \int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right] \\&\quad = 2e^{C\Gamma }C_0e^{-\delta _g\tau _0}e^{\frac{4\delta _g}{5}\tau _0}\le e^{2C\Gamma }e^{-\frac{\delta _g}{10}\tau _0}\le 1, \end{aligned}$$

which concludes the proof of (8.23) and (8.15) for \(\tau \in [\tau _0,\tau _0+\Gamma ]\).

Step 3 Finite speed of propagation. We now pick a time \(\tau _f\in [\tau _0+\Gamma ,\tau ^*]\) and propagate the bound (8.1) to the compact set \(Z\le {Z_0}\) using a finite speed of propagation argument. We claim:

$$\begin{aligned} \Vert \rho \Vert ^2_{H^{2k_0}(Z\le \frac{Z_0}{2})}+\Vert \nabla \Psi \Vert ^2_{H^{2k_0}(Z\le \frac{Z_0}{2})}\le Ce^{-{\delta _g}\tau }. \end{aligned}$$
(8.24)

Here the key is that (8.1) controls a norm on the set strictly including the light cone \(Z\le Z_2\). Let

$$\begin{aligned} \hat{Z_a}=\frac{\tilde{Z_a}+Z_2}{2} \end{aligned}$$

and note that we may, without loss of generality by taking \(a>0\) small enough, assume:

$$\begin{aligned} \frac{{\tilde{Z}}_a}{\hat{Z_a}}\le 2. \end{aligned}$$
(8.25)

Recall that \(\Gamma =\Gamma (Z_0)\) is parametrized by (8.22). We define

$$\begin{aligned} \chi (\tau ,Z)=\zeta \left( \frac{Z}{\nu (\tau )}\right) , \ \ \nu (\tau )=\frac{Z_0}{2{\hat{Z}}_a}e^{-\omega _0(\tau _f-\tau )} \end{aligned}$$

with \(\omega _0>0\) defined in (8.21), (8.22) and a fixed spherically symmetric non-increasing cut off function

$$\begin{aligned} \zeta (Z)=\left| \begin{array}{ll} 1\ \ \text{ for }\ \ 0\le Z\le {\hat{Z_a}}\\ 0\ \ \text{ for }\ \ Z\ge {\tilde{Z_a}}. \end{array}\right. , \ \ \zeta '\le 0. \end{aligned}$$
(8.26)

We define

$$\begin{aligned} \tau _\Gamma =\tau _f-\Gamma \end{aligned}$$

so that from (8.22):

$$\begin{aligned} \left| \begin{array}{l} \tau _0\le \tau _\Gamma \le \tau ^*,\\ \nu (\tau _\Gamma )=\frac{Z_0}{2{\hat{Z}}_a}e^{-\omega _0(\tau _f-\tau _\Gamma )}=\frac{Z_0}{2\hat{Z_a}}e^{-\omega _0\Gamma }=1. \end{array}\right. \end{aligned}$$
(8.27)

We pick

$$\begin{aligned} 0\le m\le 2k_0 \end{aligned}$$

then (8.26), (8.27) ensure \(\mathrm{Supp}(\chi (\tau _\Gamma ,\cdot ))\subset \{Z\le {{\tilde{Z}}_a}\}\) and hence from (8.1):

$$\begin{aligned} \left( \int (p-1)Q\rho _m^2\chi +|\nabla \Phi _m|^2\chi \right) (\tau _\Gamma )\lesssim e^{-\delta _g \tau _\Gamma }. \end{aligned}$$
(8.28)

This estimate implies that we can integrate energy identity (8.19) only on the interval \([\tau _\Gamma ,\tau _f]\). We now estimate all terms in (8.19).

Boundary terms. We compute the quadratic terms involving \(\Lambda \chi \) which should be thought of as boundary terms. First

$$\begin{aligned} \partial _\tau \chi (\tau ,Z)=-\frac{\partial _\tau \nu }{\nu } \frac{Z}{\nu }\partial _Z\zeta \left( \frac{Z}{\nu }\right) =-\omega _0\Lambda \chi . \end{aligned}$$

We now assume, recalling (8.16), that \(\omega _0\) has been chosen small enough so that (8.21) holds, and hence the lower bound on the full boundary quadratic form using \(\Lambda \chi \le 0\):

$$\begin{aligned}&\frac{1}{2}\int (p-1)Q\rho _m^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] \\&\qquad +\frac{1}{2}\int |\nabla \Phi _m|^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] +\int (p-1)Q\rho _m\nabla \chi \cdot \nabla \Phi _m\\&\quad = \int \left\{ \frac{1}{2} (p-1)Q\rho _m^2\left[ -\omega _0+H_2\right] \right. \\&\qquad \left. +\frac{1}{2}|\nabla \Phi _m|^2\left[ -\omega _0+H_2\right] +(p-1)\frac{Q}{Z}\partial _Z \Phi _m\rho _m\right\} \Lambda \chi . \end{aligned}$$

From (3.18), the discriminant of the above quadratic form is given by

$$\begin{aligned}&\left[ (p-1)\frac{Q}{Z}\right] ^2-(-\omega _0+H_2)^2(p-1)Q\\&\quad =(p-1)Q\left[ \frac{(p-1)Q}{Z^2}-(-\omega _0+H_2)^2\right] \\&\quad = (p-1)\mu ^2Q\left[ \sigma ^2-\left( -\frac{\omega _0}{\mu }+1-w\right) ^2\right] \\&\quad = (p-1)\mu ^2Q\left[ -D(Z)+O(\omega _0)\right] . \end{aligned}$$

We then observe by definition of \(\chi \) that for \(\tau \ge \tau _\Gamma \):

$$\begin{aligned} Z\in \mathrm{Supp}\Lambda \chi \Leftrightarrow {\hat{Z}}_a\le \frac{Z}{\nu (\tau )}\le {\tilde{Z}}_a \Rightarrow Z\ge \nu (\tau ) {\hat{Z}}_a\ge \nu (\tau _\Gamma ) {\hat{Z}}_a={\hat{Z}}_a \end{aligned}$$

from which since \({\hat{Z}}_a>Z_2\):

$$\begin{aligned} Z\in \mathrm{Supp}\Lambda \chi \Rightarrow -D(Z)+O(\omega _0)<0 \end{aligned}$$

provided \(0<\omega _0\ll 1\) has been chosen small enough.

Together with (8.21) and \(\Lambda \chi <0\), this ensures: \(\forall \tau \in [\tau _\Gamma ,\tau ^*]\),

$$\begin{aligned}&\frac{1}{2}\int (p-1)Q\rho _m^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] \nonumber \\&\quad +\frac{1}{2}\int |\nabla \Phi _m|^2\left[ \partial _\tau \chi +H_2\Lambda \chi \right] +\int (p-1)Q\rho _m\nabla \chi \cdot \nabla \Phi _m <0\nonumber \\ \end{aligned}$$
(8.29)

Nonlinear terms. From (8.26), (8.25) for \(\tau \le \tau _f\):

$$\begin{aligned} \text{ Supp }\chi \subset \{Z\le \nu (\tau ){\tilde{Z}}_a\}\subset \{Z\le \nu (\tau _f){\tilde{Z}}_a\}= \left\{ Z\le \frac{Z_0}{2}\frac{{\tilde{Z}}_a}{{\hat{Z}}_a}\right\} \subset \{Z\le Z_0\}, \end{aligned}$$

and hence from (8.14):

$$\begin{aligned}&\int \chi |\nabla \partial ^mG_\Phi |^2+\int (p-1)Q\chi |\partial ^mG_\rho |^2\\&\quad \lesssim \Vert \nabla G_\Phi \Vert ^2_{H^{2k_0}(Z\le Z_0)} +\Vert G_\rho \Vert ^2_{H^{2k_0}(Z\le Z_0)}\le e^{-\frac{4\delta _g}{3}\tau }. \end{aligned}$$

Conclusion. Injecting the collection of above bounds into (8.19) and summing over \(m\in [0,2k_0]\) yields the crude bound: \(\forall \tau \in [\tau _\Gamma ,\tau _f]\),

$$\begin{aligned}&\frac{d}{d\tau }\left\{ \sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi +|\nabla \Phi _m|^2\chi \right\} \\&\quad \le C \sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi +|\nabla \Phi _m|^2\chi +e^{-\frac{4\delta _g}{3}\tau }. \end{aligned}$$

We integrate the above on \([\tau _\Gamma ,\tau _f]\) and conclude using

$$\begin{aligned} \chi (\tau _f,Z)=\zeta \left( \frac{Z}{\nu (\tau _f)}\right) =\zeta \left( \frac{Z}{\frac{Z_0}{2{\hat{Z}}_a}}\right) =1\ \ \text{ for }\ \ Z\le {Z_0} \end{aligned}$$

and the initialization (8.28):

$$\begin{aligned}&\left[ \Vert \rho \Vert ^2_{H^{2k_0}(Z\le {Z_0})}+\Vert \nabla \Psi \Vert ^2_{H^{2k_0}(Z\le {Z_0})}\right] (\tau _f)\\&\quad \lesssim e^{C(\tau _f-\tau _\Gamma )}e^{-\delta _g\tau _\Gamma }+\int _{\tau _\Gamma }^{\tau _f}e^{C(\tau _f-\sigma )}e^{-\frac{4\delta _g}{3}\sigma }d\sigma \\&\quad \lesssim C(\Gamma )e^{-\delta _g\tau _f}=C(Z_0)e^{-\delta _g\tau _f}. \end{aligned}$$

Since the time \(\tau _f\) is arbitrary in \([\tau _0+\Gamma ,\tau ^*]\), the bound (8.24) follows.

Step 4 Proof of (8.15). We run the energy identity (8.19) with \(\chi _{\nu _0,m}\) and estimate each term.

Terms \(\frac{Z_0}{3}\le Z\le \frac{Z_0}{2}\). In this zone, we have by construction

$$\begin{aligned} \rho ={\tilde{\rho }}\end{aligned}$$

and hence the bootstrap bounds (4.38) imply

$$\begin{aligned} \Vert \rho \Vert _{H^{k_m}(Z\le \frac{Z_0}{2})}+\Vert \nabla \Psi \Vert _{H^{k_m}(Z\le \frac{Z_0}{2})}\lesssim 1 \end{aligned}$$

and hence interpolating with (8.24) for \(k_m \) large enough:

$$\begin{aligned} \Vert \rho \Vert _{H^{m}(\frac{Z_0}{3}\le Z\le \frac{Z_0}{2})}\lesssim & {} \Vert \rho \Vert ^{\frac{m}{k_m}}_{H^{k_m}(\frac{Z_0}{3}\le Z\le \frac{Z_0}{2})}\Vert \rho \Vert ^{1-\frac{m}{k_m}}_{L^2(\frac{Z_0}{3}\le Z\le \frac{Z_0}{2})}\lesssim e^{-\frac{\delta _g}{2}\left( 1-\frac{m}{k_m}\right) } \nonumber \\\le & {} e^{-\frac{4\delta _g}{10}} \end{aligned}$$
(8.30)

and similarly for the phase

$$\begin{aligned} \Vert \nabla \Psi \Vert _{H^{m}(\frac{Z_0}{3}\le Z\le \frac{Z_0}{2})}\lesssim e^{-\frac{\delta _g}{2}\left( 1-\frac{m}{k_m}\right) }\le e^{-\frac{4\delta _g}{10}}. \end{aligned}$$
(8.31)

Linear term. We observe the cancellation using (8.17), (4.2):

$$\begin{aligned} \partial _\tau \chi _{\nu _0,m}+H_2\Lambda \chi _{\nu _0,m}= & {} \frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)}}\left[ -\mu \Lambda \zeta \left( \frac{Z}{Z^*}\right) \right] \nonumber \\&+ \mu (1-w)\left[ \frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)}}\Lambda \zeta \left( \frac{Z}{Z^*}\right) \right. \nonumber \\&\left. +\Lambda \left( \frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)}}\right) \zeta \left( \frac{Z}{Z^*}\right) \right] \nonumber \\= & {} -\mu \left[ d-2(r-1)+2(\nu _0-m)\right] \chi _{\nu _0,m} \nonumber \\&+O\left( \frac{1}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)+\omega }}\right) \end{aligned}$$
(8.32)

for some universal constant \(\omega >0\). We now estimate the norm for \(2Z^*\le Z\le 3Z^*\). Using spherical symmetry for \(Z\ge 1\) and \(m\ge 1\):

$$\begin{aligned} |Z^{m}\partial ^m\rho |\lesssim \sum _{j=1}^mZ^{m}\frac{|\partial _Z^j\rho |}{Z^{m-j}}\lesssim \sum _{j=1}^m Z^{j}|\partial _Z^j\rho | \end{aligned}$$
(8.33)

and hence using the outer \(L^\infty \) bound (4.40):

$$\begin{aligned}&\int _{2Z^*\le Z\le 3Z^*}\frac{(p-1)Q|\partial ^m\rho |^2+|\partial ^m\nabla \Phi |^2}{\langle Z\rangle ^{d-2(r-1)+2(\nu _0-m)+\omega }}\nonumber \\&\quad \lesssim \int _{2Z^*\le Z\le 3Z^*}\left[ \sum _{j=0}^m \left| \frac{Z^{j}\partial _Z^j\rho }{\langle Z\rangle ^{\frac{d}{2}+\nu _0+\frac{\omega }{2}}}\right| ^2+\sum _{j=1}^{m+1}\left| \frac{Z^{j}\partial _Z^j\Phi }{\langle Z\rangle ^{\nu _0+\frac{d}{2}-(r-1)+1+\frac{\omega }{2}}}\right| ^2\right] \nonumber \\&\quad \lesssim \int _{2Z^*\le Z\le 3Z^*}\left[ \sum _{j=0}^m \left| \frac{Z^{j}\partial _Z^j\rho }{\rho _P\langle Z\rangle ^{\frac{d}{2}+\nu _0+\frac{2(r-1)}{p-1}+\frac{\omega }{2}}}\right| ^2 \right. \nonumber \\&\qquad \left. +\sum _{j=1}^{m+1}\left| \langle Z\rangle ^{r-2}\frac{Z^{j}\partial _Z^j\Psi }{\langle Z\rangle ^{\nu _0+\frac{2(r-1)}{p-1}+\frac{d}{2}+\frac{\omega }{2}}}\right| ^2\right] \nonumber \\&\quad \lesssim \frac{1}{(Z^*)^{\omega +2\left[ \nu _0+\frac{2(r-1)}{p-1}\right] }}\left( 1+b^2(Z^*)^{2(r-2)}\right) \le e^{-\delta _g\tau } \end{aligned}$$
(8.34)

using

$$\begin{aligned} b(Z^*)^{r-2}=e^{\tau \left[ -{\mathscr {e}}+\mu (r-2)\right] }=e^{\tau \left[ -{\mathscr {e}}+1-2\mu \right] }=1 \end{aligned}$$

and the explicit choice from (4.17):

$$\begin{aligned} 2\mu \left( \nu _0+\frac{2(r-1)}{p-1}\right) =\delta _g. \end{aligned}$$

Conclusion Injecting the above bounds into (8.19) yields:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right\} \\&\quad =-\int \chi _{\nu _0,m}\left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \left[ \mu \nu _0+\frac{2\mu (r-1)}{p-1}\right] \\&\qquad + O\left( \int _{Z_0\le Z\le 2Z^*} \chi _{\nu _0,m}\left[ \sum _{m=0}^{m+1}\frac{|\partial _Z^j\Phi |^2}{\langle Z\rangle ^{2(m+1-j)+2\omega }}\right. \right. \\&\qquad \left. \left. +\sum _{j=0}^{m}\frac{Q|\partial _Z^j\rho |^2}{\langle Z\rangle ^{2(m-j)+2\omega }}\right] +e^{-\frac{4\delta _g}{5}\tau }\right) \\&\qquad + O\left( \int \chi _{\nu _0,m}|\nabla \Phi _m||\nabla \partial ^mG_\Phi |+\int \chi _{\nu _0,m}Q|\rho _m||\partial ^mG_\rho |\right) \end{aligned}$$

and hence after summing over m:

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right\} \\&\quad = -\mu \left[ \nu _0+\frac{2(r-1)}{p-1}\right] \sum _{m=0}^{2k_0} \int \chi _{\nu _0,m}\left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \\&\qquad + O\left( e^{-\frac{4\delta _g}{5}\tau }+\sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi _{\nu _0+\omega ,m}+|\nabla \Phi _m|^2\chi _{\nu _0+\omega ,m}\right) \\&\qquad + \sum _{m=0}^{2k_0}O\left( \int \chi _{\nu _0,m}|\nabla \Phi _m||\nabla \partial ^mG_\Phi |+\int \chi _{\nu _0,m}Q|\rho _m||\partial ^mG_\rho |\right) \end{aligned}$$

Using (8.24) we conclude

$$\begin{aligned}&\frac{1}{2}\frac{d}{d\tau }\left\{ \sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right\} \nonumber \\&\quad = -\mu \left[ \nu _0+\frac{2(r-1)}{p-1}+O\left( \frac{1}{Z_0^C}\right) \right] \nonumber \\&\qquad \times \sum _{m=0}^{2k_0} \int \chi _{\nu _0,m}\left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] \nonumber \\&\qquad + O\left( e^{-\frac{4\delta _g}{5}\tau }+\sum _{m=0}^{2k_0} \int \chi _{\nu _0,m}|\nabla \partial ^mG_\Phi |^2\right. \nonumber \\&\qquad \left. +\int (p-1)Q\chi _{\nu _0,m}|\partial ^mG_\rho |^2\right) . \end{aligned}$$
(8.35)

Therefore, using also (8.20), for \(Z_0\) large enough and universal and

$$\begin{aligned} 2\mu \left( \nu _0+\frac{2(r-1)}{p-1}\right) =\delta _g, \end{aligned}$$

there holds

$$\begin{aligned}&\frac{d}{d\tau }\left\{ \sum _{m=0}^{2k_0}\int (p-1)Q\rho _m^2\chi _{\nu _0,m}+|\nabla \Phi _m|^2\chi _{\nu _0,m}\right\} \\&\quad \le -\frac{4\delta _g}{5}\sum _{m=0}^{2k_0} \int \chi _{\nu _0,m}\left[ (p-1)Q\rho _m^2+|\nabla \Phi _m|^2\right] +C e^{-\frac{4\delta _g\tau }{5}}. \end{aligned}$$

Integrating in time and using (4.19) yields (8.15). \(\square \)

8.3 Closing the bootstrap and proof of Theorem 1.1

We are now in position to prove the bootstrap Proposition 4.4 which immediately implies Theorem 1.1.

Proof of Proposition 4.4 and Theorem 1.1

Recall that the non vanishing of the solution is ensured by (4.27). It remains to close the bound (4.26). Indeed, from (4.1), (4.2), (4.8) for \(Z\ge Z^*\):

$$\begin{aligned} \frac{|\Delta u|}{\rho _D}\lesssim \frac{(Z^*)^2}{\rho _D}\left[ |\Delta \rho _{\mathrm{Tot}}|+\frac{|\partial _Z\rho _{\mathrm{Tot}}||\partial _Z\Psi _{\mathrm{Tot}}|}{b}+\frac{|\rho _{\mathrm{Tot}}\Delta \Psi _{\mathrm{Tot}}|}{b}\right] \lesssim 1, \end{aligned}$$

where we used (4.40) in the last step. The \(|u|^p\) term is handled similarily, and (4.26) is improved for \(b_0\) small enough.Footnote 14 Note also that the bounds (4.40) imply

$$\begin{aligned} \Vert u(t)\Vert _{H^{k_c}}\le C(t) \end{aligned}$$

for \(\frac{d}{2}\ll k_c\ll k_m\) for times in the bootstrap interval and hence the bootstrap time is strictly smaller than the life time provided by standard Cauchy theory.

We now conclude from a classical topological argument à la Brouwer. The bounds of Sects. 5678 have been shown to hold for all initial data on the time interval \([\tau _0, \tau _0+\Gamma ]\) with \(\Gamma \) large. Moreover, as explained in the proof of Lemma 8.1, they can be immediately propagated to any time \(\tau ^*\) after a choice of projection of initial data on the subspace of unstable modes \({{\mathbb {P}}}X(\tau _0)\).

This is done as follows. We define a decomposition of the the set of initial data \(X(\tau _0)\). Recall that the restriction of the data \(X(\tau _0)\) to the interval \([0,Z_a]\) is contained in the Sobolev space \({{\mathbb {H}}}_{2k_0}\) which can be split into a direct sum of the stable and unstable subspaces

$$\begin{aligned} {{\mathbb {H}}}_{2k_0}=U\bigoplus V. \end{aligned}$$

For functions defined for all Z, we define a subspace of functions satisfying the assumptions of Sect. 4.3 on initial data

$$\begin{aligned} V_{reg}=\{({{\tilde{\rho }}},\Psi ): \,\Vert {{\tilde{\rho }}},\Psi \Vert _{k_m}<\infty \} \end{aligned}$$

with the property that \(V_{reg}\) has the same dimension as V and its restriction to \([0,Z_a]\) satisfies the property

$$\begin{aligned} {\text {dist}} (V_{reg}, V)_{{{\mathbb {H}}}_{2k_0}} < e^{-3\delta _g \tau _0}. \end{aligned}$$
(8.36)

Note that the space \(V_{reg}\) consists of functions which are defined for all Z and which are more regular on \([0,Z_a]\) than the ones contained in V. We can explicitly construct \(V_{reg}\) by defining it as the linear space generated by \(\{v^1_{reg},\ldots ,v^n_{reg}\}\) where each \(v^j_{reg}\) is obtained from the element \(v^j\)—a generator of V—by a smoothing and an extension procedure. The precise details of both the smoothing and extension are not important, as long as (8.36) is ensured to hold. By (8.36), the projection \({{\mathbb {P}}}\) (composed with the restriction to \([0,Z_a]\)) is an isomorphism between \(V_{reg}\) and V. Denoting the inverse of this isomorphism by \({{\mathbb {I}}}\), we see that it satisfies the property that \({{\mathbb {P}}}\circ {{\mathbb {I}}}\) is the identity map on V. We also define a complementary subspace W such that the space of all data with \(\Vert {{\tilde{\rho }}},\Psi \Vert _{k_m}<\infty \) decomposes into the sum

$$\begin{aligned} V_{reg}\bigoplus W \end{aligned}$$

with W obeying the additional property that \({{\mathbb {P}}} W=0\). We can further restrict W to consist of functions satisfying all of the assumptions of Sect. 4.3 on initial data. Let \(X(\tau )\) be the solution of the nonlinear problem (3.14) with the initial data \(X(\tau _0)\). We now apply Lemma 3.5 to \(x(\tau )={{\mathbb {P}}}X(\tau )\). We choose the initial data \(X(\tau _0)\) obeying all the initial data bounds with the additional condition that it is of the form \(w+{{{\mathbb {I}}}} v\) with a fixed element \(w\in W\) obeying the bound

$$\begin{aligned} \Vert w\Vert _{{{\mathbb {H}}}_{2k_0}}\le e^{-\frac{\delta _g}{2}\tau _0} \end{aligned}$$

and any element \(v\in V\) obeying

$$\begin{aligned} \Vert v\Vert _{{{\mathbb {H}}}_{2k_0}}\le e^{-\frac{3\delta _g}{5}\tau _0}. \end{aligned}$$

For such initial data \({{\mathbb {P}}} X(\tau _0)=v\) and the solution \(X(\tau )\) depends continuously on v. As a consequence, the right hand side F of Lemma 3.5, which represents the projection \({{\mathbb {P}}}\) of the nonlinear terms G in (3.14), can be shown to depend continuously on \(x(\tau )\). All other assumptions of Lemma 3.5 follow from the bounds of Sects. 5678 and Lemma 8.1. It now follows from Lemma 3.5 that, for any w fixed as above, there exists \(v^*=v^*(w)\) such that the exit time \(\tau ^*\) corresponding to the initial data \(X(\tau _0)={{\mathbb {I}}} v^*+w\) satisfies \(\tau ^*=\infty \). This concludes the proof of Theorem 1.1. \(\square \)