Shadows of infinities

Abstract

We study unbounded “supersolutions” of the Evolutionary p-Laplace equation with slow diffusion. They are the same functions as the viscosity supersolutions. A fascinating dichotomy prevails: either they are locally summable to the power $p-1+\tfrac{n}{p}-0$ or not summable to the power $p-2+0$.

1 Introduction

Our object is the unbounded supersolutions of the Evolutionary p-Laplace equation

$$\begin{aligned} \frac{\partial v}{\partial t}\,=\;\nabla \! \cdot \! \left( |\nabla v|^{p-2}\nabla v\right) , \quad 2 < p < \infty , \end{aligned}$$

(1)

in a domain $\Omega _T = \Omega \times (0,T),$ where $\Omega$ is a connected open domain in $\mathbb {R}^n$. Here $v = v(x_1,x_2,\ldots ,x_n,t)$ and $\nabla v = (\tfrac{\partial v}{\partial x_1},\tfrac{\partial v}{\partial x_2},\ldots ,\tfrac{\partial v}{\partial x_n})$. In the literature supersolutions are usually treated as weak supersolutions to the equation, but we are interested in a much wider class of functions. Our “supersolutions” are defined at each point in $\Omega _T,$ are lower semicontinuous, and obey a comparison principle with respect to the solutions of the equation (There is no assumption about $\nabla v$). If such a supersolution, in addition, is finite in a dense subset of $\Omega _T$, we call it a p-superparabolic function.¹ The p-superparabolic functions now defined as in Potential Theory are, incidentally, the same functions as the viscosity supersolutions of the Evolutionary p-Laplace equation, cf. [7]. They appear in obstacle problems and are relevant for the Perron method, see [8].

The three cases $1 < p < 2$ (fast diffusion), $p = 2$ (the Heat Equation) and $2 < p < \infty$ (slow diffusion) are very different. We shall treat only the slow diffusion case $p > 2$. In this case disturbances propagate, as it were, with finite speed. But, as we shall see, “infinite values” seem to propagate with infinite speed. We have detected some fascinating phenomena, which are totally absent from the linear theory.

We are interested in the set of points where “$v(x,t) = \infty$”, the so-called infinities (We do not want to call them poles). Their definition is delicate. There are several possibilities, but, first of all, the right definition must agree with the concept in the stationary case $\nabla \!\cdot \! (|\nabla u|^{p-2}\nabla u),\; u = u(x)$. The following two sets of infinities

$$\begin{aligned} \Xi ^{\perp }&= \left\{ (x_0,t_0)|\,\underset{(x,t)\rightarrow (x_0,t_0+)}{\lim }v(x,t) = + \infty \right\} \\ \Xi ^{\downarrow }&= \left\{ (x_0,t_0)|\,\underset{t \rightarrow t_0+}{\lim }v(x_0,t) = + \infty \right\} \end{aligned}$$

are of interest for a p-superparabolic function v,  but in principle one could consider any set $\Xi$ such that $\Xi ^{\perp } \subset \Xi \subset \Xi ^{\downarrow }$. In $\Xi ^{\perp }$ the limit is taken via neighborhoods of the type

$$\begin{aligned} |x-x_0| < \rho ,\quad t_0 < t < t_0 + \delta . \end{aligned}$$

In $\Xi ^{\downarrow }$ only the time variable moves. It is of utmost importance that the limits are determined only by the future times $t > t_0,$ while the past and present times $t \le t_0$ are totally excluded from the definitions of $\Xi ^{\perp }$ and $\Xi ^{\downarrow }$. This is in striking contrast to the actual pointwise value of the function, which can always be determined by only the past:

$$\begin{aligned} v(x_0,t_0) = \liminf _{\begin{array}{c} (x,t) \rightarrow (x_0,t_0)\\ t < t_0 \end{array}} v(x,t). \end{aligned}$$

See [9]. Therefore, it may so happen that $v(x_0,t_0) < \infty ,$ although $(x_0,t_0) \in \Xi ^{\perp }$. This feature is not easily dismissed. Nonetheless, we call $(x_0,t_0)$ a point of infinity for v, or, just an infinity (In the stationary case they are poles).

At this stage, we interrupt our tale by introducing the celebrated Barenblatt solution

$$\begin{aligned} \mathfrak {B}(x,t) = {\left\{ \begin{array}{ll} t^{-\tfrac{n}{\lambda }}\left[ C-\frac{p-2}{p}\lambda ^{\tfrac{1}{1-p}}\left( \frac{|x|}{t^{1/\lambda }}\right) ^{\frac{p}{p-1}}\right] _{+}^{\tfrac{p-1}{p-2}}, &{} \text {when} \quad t > 0\\ 0, &{} \text {when} \quad t \le 0 \end{array}\right. } \end{aligned}$$

(2)

found in 1951, cf. [1]. Here $\lambda = n(p-2)+p$. It is a solution of the Evolutionary p-Laplace Equation, except at the origin $x=0, t=0$. It is a p-superparabolic function in the whole $\mathbb {R}^n\times \mathbb {R},$ where it satisfies the equation

$$\begin{aligned} \frac{\partial \mathfrak {B}}{\partial t} - \nabla \cdot (|\nabla \mathfrak {B}|^{p-2}\nabla \mathfrak {B}) = c \delta \end{aligned}$$

in the sense of distributions ($\delta =$ Dirac’s delta). Note carefully that due to the requirement of semicontinuity, it follows that $\mathfrak {B}(0,0) = 0$ and not $=\infty$ at the point $(0,0)\in \Xi ^{\downarrow }$. Note however that $(0,0) \not \in \Xi ^{\perp }$. In passing, we cannot resist mentioning that even for the Heat equation

$$\begin{aligned} \frac{\partial v}{\partial t} = \Delta v \end{aligned}$$

a similar situation appears with the fundamental solution

$$\begin{aligned} \mathfrak {W}(x,t) = {\left\{ \begin{array}{ll} \dfrac{1}{(4\pi t)^{\frac{n}{2}}}e^{-\frac{|x|^2}{4t}},&{} \text {when} \quad t > 0\\ 0, &{} \text {when} \quad t \le 0. \end{array}\right. } \end{aligned}$$

Now $\mathfrak {W}(0,0) = 0$ while $\lim _{t \rightarrow 0+}\mathfrak {W}(0,t) = \infty$. In classical Potential Theory, one often introduces an auxiliary supercaloric function in order to include $\{(0,0)\}$ among the “polar sets”, see [23]. Such an awkward procedure is not natural for $p > 2$. In the nonlinear theory the presence of the original p-superparabolic function is central.

In order to proceed, we recall that the p-superparabolic functions were required to be finite in a dense subset. Although this at least excludes “supersolutions” that are identically infinite during some time interval, arbitrarily fast growth is still possible. For example, there are p-superparabolic functions of the form

$$\begin{aligned} v(x,t) = {\left\{ \begin{array}{ll} u(x)e^{+\frac{1}{(p-2)(t-t_0)}}, &{} \text {when} \quad t > t_0\\ 0, &{} \text {when} \quad t \le t_0 \end{array}\right. } \end{aligned}$$

where $u(x) > 0$ in $\Omega$. Notice that here the set $\Xi ^{\perp } = \Omega \times \{t_0\}$. As we shall see, the property that the infinities occupy the whole $\Omega$ at some time $t_0$ is a typical phenomenon for a class of p-superparabolic functions.

An important result is that the p-superparabolic functions $v:\,\Omega _T \rightarrow (-\infty ,\infty ]$ are of two different kinds:

• Class $\mathfrak {B}$       $v \in L_{loc}^{p-1+\frac{p}{n}-\varepsilon }(\Omega _T)$    for each $\varepsilon > 0$

• Class $\mathfrak {M}$       $v \not \in L_{loc}^{p-2+\varepsilon }(\Omega _T)$    for each $\varepsilon > 0$.

There is no third possibility.² The classes are not empty. The void gap

$$\begin{aligned} \bigl (p-2,\,p-1+\frac{p}{n}\bigr ) \end{aligned}$$

is remarkable, to say the least. In other words, if $v \in L_{loc}^s(\Omega _T)$ for some $s > p-2,$ then $v \in L_{loc}^q(\Omega _T)$ whenever $q < p-1+\tfrac{p}{n}$ (Lemma 12). The functions of class $\mathfrak {B}$ have the important property³ that their gradients $\nabla v$ exist in Sobolev’s sense and

$$\begin{aligned} \nabla v \in L_{loc}^q(\Omega _T)\quad \text {whenever} \quad q < p-1+ \frac{1}{n+1} \end{aligned}$$

(Theorem 13). As a consequence, there exists a Radon measure $\mu \ge 0$ depending, of course, on v such that

$$\begin{aligned} \frac{\partial v}{\partial t} - \nabla \! \cdot \! \left( |\nabla v|^{p-2}\nabla v\right) = \mu \end{aligned}$$

(3)

in the sense of distributions. The p-superparabolic functions of class $\mathfrak {B}$ have a well-established theory, described in [2] for example. See also [11].

The functions in class $\mathfrak {M}$ seem to have few good properties. First, they do not induce a Radon measure. Second, strictly speaking, their Sobolev derivative $\nabla v$ does not exist. Thus it is important to achieve simple criteria to detect functions of class $\mathfrak {M}$. Fortunately, their sets of infinities always contain a whole time slice $t=t_0,$ i.e., $v(x,t_0+) \equiv \infty$ when $x \in \Omega$. This cannot happen for the $\mathfrak {B}$-class. The following criterion also assures that if there are too many infinities inside the domain at the same time, they have to touch the lateral boundary. They cast their shadows on the boundary.

Theorem 1

(Theorema Infinitorum) A p-superparabolic function is of class $\mathfrak {M}$ if and only if there is a time $t_0$ such that

$$\begin{aligned} \Xi ^{\downarrow }(t_0) \equiv \left\{ (x,t_0)\in \Xi ^{\downarrow }\right\} = \Omega \times \{t_0\}. \end{aligned}$$

Further, if $\Xi ^{\downarrow }(t_0)$ has positive n-dimensional Lebesgue measure,⁴ then $\Xi ^{\downarrow }(t_0) = \Omega \times \{t_0\}$. The same holds for $\Xi ^{\perp }(t_0)$.

Recall that always $\Xi ^{\perp }(t_0) \subset \Xi ^{\downarrow }(t_0)$. A peculiarity, which we have found, appears when $\Omega$ is the whole space $\mathbb {R}^n$. In class $\mathfrak {M}$ there are no non-negative p-superparabolic functions defined in $\mathbb {R}^n\times (0,T),$ see Theorem 19. At first sight, their absence is surprising.

Our method is based on the bounded p-superparabolic functions

$$\begin{aligned} v_k = v_k(x,t) = \min \{v(x,t),k\}. \end{aligned}$$

Bounded p-superparabolic functions belong to the natural Sobolev space $L^p_{loc}(0,T,W^{1,p}_{loc}(\Omega ))$ and are weak supersolutions of the Eq. (1), cf. [9, 17]. Thus a priori estimates like the Caccioppoli inequalities are available. A convenient version of an inequality of Harnack’s type, given in [14], is needed in our work. It is valid for non-negative p-superparabolic functions that are bounded, in particular for the above $v_k$’s (see Lemma 10 below).

Let us first mention a few selected results for class $\mathfrak {B}$:

Theorem 2

(Class $\mathfrak {B}$) For a p-superparabolic function $v:\, \Omega _T \rightarrow (- \infty ,\infty ]$ the following conditions are equivalent:

• (i)    $v \in L_{loc}^s(\Omega _T)\qquad \text {for some}\quad s > p-2$.

• (ii)    $\nabla v \quad \text {exists and} \quad \nabla v \in L_{loc}^q(\Omega _T) \quad \text {whenever} \quad q < p-1+ \frac{1}{n+1}$.

• (iii)    When $\delta >0,$

  $$\begin{aligned} \underset{0<t<T-\delta }{{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\int _{D}\!|v(x,t)|\,\,\mathrm {d}x< \infty ,\quad \text {if}\quad D \subset \subset \Omega . \end{aligned}$$

• (iv)    The n-dimensional measure $|\Xi ^{\downarrow }(t)| = 0$ for each $0 < t < T$

• (v)    It never happens that

  $$\begin{aligned} \lim _{\begin{array}{c} (y,t) \rightarrow (x,t_0)\\ t>t_0 \end{array}} v(y,t) = \infty \quad \text {for all}\quad x \in \Omega . \end{aligned}$$

Condition (ii) has the important implication that there exists a non-negative Radon measure $\mu$, depending on v,  such that the equation

$$\begin{aligned} \int _{0}^{T}\!\!\int _{\Omega }\Bigl (-v\frac{\partial \varphi }{\partial t}+ \langle |\nabla v|^{p-2}\nabla v, \nabla \varphi \rangle \Bigr ) \,\mathrm {d}x \,\mathrm {d}t = \int _{\Omega _T}\!\varphi \,\,\mathrm {d}\mu \end{aligned}$$

(4)

holds for all test functions $\varphi \in C_{0}^{\infty }(\Omega _T)$. In other words, Eq. (3) holds in the sense of distributions, cf. [11].

Then we characterize the $\mathfrak {M}$ class:

Theorem 3

( Class $\mathfrak {M}$) For a p-superparabolic function $v:\, \Omega _T \rightarrow (- \infty ,\infty ]$ the following conditions are equivalent:

• (i)    $v \not \in L_{loc}^{p-2+\varepsilon }(\Omega _T)\quad \text {for any} \quad \varepsilon > 0$.

• (ii)    For some $\delta >0,$

  $$\begin{aligned} \underset{0<t<T-\delta }{{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\int _{D}\!|v(x,t)|\,\,\mathrm {d}x=\infty \quad \text {when}\quad D \subset \subset \Omega ,\quad |D|>0. \end{aligned}$$

• (iii)    There is $t_0$ such that the n-dimensional measure $|\Xi ^{\downarrow }(t_0)| > 0$.

• (iv)    There is $t_0$ such that $\Xi ^{\downarrow }(t_0) = \Omega \times \{t_0\}$.

• (v)    At some point $(x_0,t_0),$

  $$\begin{aligned} \liminf _{\begin{array}{c} (y,t) \rightarrow (x_0,t_0)\\ t>t_0 \end{array}} \bigl (v(y,t)(t-t_0)^{\frac{1}{p-2}}\bigr ) > 0. \end{aligned}$$

Actually, statements (i) in the previous theorems could be slightly sharpened to the form $v\in L_{loc}^{p-2}(\Omega _T)$ in Theorem 2 and $v\notin L_{loc}^{p-2}(\Omega _T)$ in Theorem 3. This follows from condition (v) in Theorem 3 (see also Lemma 11).

From this, one can read off a simple sufficient condition to guarantee that a p-superparabolic function belongs to class $\mathfrak {B}$. It is clear that a function which is bounded near the boundary cannot belong to class $\mathfrak {M}$. More precisely, if

$$\begin{aligned} \limsup _{(x,\tau )\rightarrow (\xi ,t)}v(x,\tau ) <\infty \quad \text {for all} \quad (\xi ,t) \in \partial \Omega \times [0,T) \end{aligned}$$

then v is of class $\mathfrak {B}$. A further result, which we find astonishing, is that $\Xi ^{\perp }$ cannot contain a portion with positive area of any other hyperplane intersecting $\Omega _T$ than those of the type $t=$ Const.

Proposition 4

If  $\Xi ^{\perp }$ contains a portion with positive area of the hyperplane

$$\begin{aligned} t = \langle a,x\rangle + \alpha \end{aligned}$$

then $a = \overline{0}$.

The p-superparabolic functions of class $\mathfrak {M}$ do not induce a $\sigma$-finite measure $\mu$ and are therefore beyond the scope of most articles devoted to the Evolutionary p-Laplace equation. The fatal feature is that the possibility

$$\begin{aligned} \iint _{K} \!|\nabla v|^{p-1} \,\mathrm {d}x \,\mathrm {d}t = \infty ,\quad K \subset \subset \Omega _T, \end{aligned}$$

cannot be avoided, in which case Eq. (4) does not make sense. Moreover, $\nabla v$ does not exist in Sobolev’s sense. However, if $v\ge 1$ we know from [10], Theorem 4.3, that $\nabla \log v$ exists, and the Caccioppoli estimate

$$\begin{aligned} \int _0^T\!\!\int _{\Omega }|\nabla \log v|^p\zeta ^p\,\,\mathrm {d}x \,\mathrm {d}t\le & {} c\int _0^T\!\!\int _{\Omega }v^{2-p}\Bigl \vert \frac{\partial \zeta ^p}{\partial t}\Bigr \vert \,\mathrm {d}x \,\mathrm {d}t \nonumber \\&+ c \int _0^T\!\!\int _{\Omega }|\nabla \zeta |^p\,\,\mathrm {d}x \,\mathrm {d}t \end{aligned}$$

(5)

holds.

Infinite initial values for solutions There are interesting consequences for the solutions of the Cauchy–Dirichlet problem in $\Omega _T$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u}{\partial t}\;=\;\nabla \! \cdot \! \bigl (|\nabla u|^{p-2}\nabla u \bigr )\\ u(x,0) \;=\;g(x),\quad \text {when} \quad x \in \Omega , \end{array}\right. } \end{aligned}$$

(6)

where $0\,\le g(x)\,\le \infty ,$ if infinite initial values are prescribed. The lateral boundary values are not essential now. Let us suppose that u is a weak solution in $\Omega _T$ and that $u \ge 0$ in $\Omega _T$, see Definition 5 in Sect. 2. We assume that the initial values are infinite in a set $E\;\subset \;\Omega$:

$$\begin{aligned} \lim _{t \rightarrow 0+}u(x,t)\;=\;+\infty \quad \text {for all}\quad x \in E. \end{aligned}$$

(7)

The following results come from our study:

• If the n-dimensional measure of E is strictly positive, then $E\,=\,\Omega$.

• There exist solutions, if $E\,=\,\Omega$ and $\Omega$ is bounded.

• If $\Omega \,=\, \mathbb {R}^n$ and the measure of E is positive, then there is no solution.

To spell it out, the requirement that

$$\begin{aligned} \limsup _{t\rightarrow 0+}u(x_0,t)\;< \; \infty \end{aligned}$$

at some point $x_0$ is incompatible with the condition that (7) holds in a set E of positive measure. If we replace $\Omega _T$ with a domain like

$$\begin{aligned} \Upsilon \;=\;\{(x,t)\,|x\in \Omega ,\,\Psi (x) < t < T \}, \end{aligned}$$

where $\Psi \,=\,\Psi (x)$ is a smooth function, then the corresponding initial condition

$$\begin{aligned} \lim _{t\rightarrow \Psi (x)+}u(x,t)\;=\;\infty \quad \text {at every}\quad x\in \Omega \end{aligned}$$

is impossible, except when $\Psi (x) =$ constant. Thus we are back to the space$\times$time cylinders. We hope to return to this matter in a later work.

2 Preliminaries

We begin with some standard notation. We consider an open domain $\Omega$ in $\mathbb {R}^n$ and denote by $L^p(t_1,t_2;W^{1,p}(\Omega ))$ the Sobolev space of functions $v = v(x,t)$ such that for almost every $t,\, t_1 \le t \le t_2,$ the function $x \mapsto v(x,t)$ belongs to $W^{1,p}(\Omega )$ and

$$\begin{aligned} \int _{t_1}^{t_2}\!\!\int _{\Omega }\bigl (|v(x,t)|^p + |\nabla v(x,t)|^p\bigr )\,\mathrm {d}x \,\mathrm {d}t \;< \;\infty , \end{aligned}$$

where $\nabla v = (\tfrac{\partial v}{\partial x_1},\ldots ,\tfrac{\partial v}{\partial x_2})$. The definition of the local space $L^p(t_1,t_2;W^{1,p}_{loc}(\Omega ))$ is analogous. The space $L^p_{loc}(t_1,t_2;W^{1,p}_{loc}(\Omega ))$ is also used.

Definition 5

Let $u \in L^p(t_1,t_2;W^{1,p}(\Omega ))$. Then u is a weak solution of the Evolutionary p-Laplace equation in $\Omega \times (t_1,t_2),$ if

$$\begin{aligned} \int _0^T\!\!\int _{\Omega }\Bigl (-u\frac{\partial \varphi }{\partial t} + \langle |\nabla u|^{p-2}\nabla u, \nabla \varphi \rangle \Bigr ) \,\mathrm {d}x \,\mathrm {d}t\; =\; 0 \end{aligned}$$

(8)

whenever $\varphi \in C_0^{\infty }(\Omega \times (t_1,t_2))$. If, in addition, u is continuous, then it is called a p-parabolic function. Further, we say that u is a weak supersolution, if the above integral is $\ge 0$ for all $\varphi \ge 0$ in $C_0^{\infty }(\Omega \times (t_1,t_2))$. If the integral is non-positive instead, we say that u is a weak subsolution.

By parabolic regularity theory, a weak solution is locally Hölder continuous after a possible redefinition in a set of $n\!+\!1$-dimensional Lebesgue measure zero, see [21] and [4]. Also a weak supersolution can be made semicontinuous through such a redefinition, cf. [13]. Then it is a p-superparabolic function according to the Comparison Principle below.

Lemma 6

(Comparison principle) Assume that u and v belong to $L^p\bigl (t_1,t_2;W^{1,p}(\Omega )\bigr ) \bigcap C\bigl (\overline{\Omega } \times [t_1,t_2)\bigr )$. If v is a weak supersolution and u a weak subsolution in $\Omega _{t_1,t_2} = \Omega \times (t_1,t_2)$ such that

$$\begin{aligned} v \ge u \quad \text {on the parabolic boundary} \quad \overline{\Omega } \times \{t_1\} \bigcup \partial \Omega \times (t_1,t_2), \end{aligned}$$

then $v \ge u$ in the whole $\Omega _{t_1,t_2}$.

The Comparison Principle is used to define p-superparabolic functions:

Definition 7

A function $v:\,\Omega \times (t_1,t_2)\mapsto (-\infty ,\infty ]$ is called a p-superparabolic function if the conditions

• v is lower semicontinuous

• v is finite in a dense subset

• v satisfies the comparison principle on each cylinder $D_{t'_1,t'_2} = D \times (t'_1,t'_2) \subset \subset \Omega _{t_1,t_2}$: if $h \in C(\overline{D_{t'_1,t'_2}})$ is a p-parabolic function in $D_{t'_1,t'_2},$ and if $h \le v$ on the parabolic boundary of $D_{t'_1,t'_2}$, then $h \le v$ in the whole $D_{t'_1,t'_2}$

are valid.

We recall a fundamental result from [10], Theorem 1.4; see [17] for a better proof based on infimal convolutions. See also [12].

Theorem 8

Let $p \ge 2$. If v is a p-superparabolic function that is locally bounded from above in $\Omega _T,$ then the Sobolev gradient $\nabla v$ exists and $\nabla v \in L^p_{loc}(\Omega _T)$. Moreover, v is a weak supersolution.

In order to derive estimates from the theorem, we need bounded functions. The truncations $v_k = \min \{v(x,t),k\}$ are p-superparabolic, if v is, and they are bounded from above. Thus $\nabla v_k$ is at our disposal and estimates derived from the inequality

$$\begin{aligned} \int _0^T\!\!\int _{\Omega }\Bigl (-v_k\frac{\partial \varphi }{\partial t} + \langle |\nabla v_k|^{p-2}\nabla v_k, \nabla \varphi \rangle \Bigr )\! \,\mathrm {d}x \,\mathrm {d}t \;\ge \;0 \end{aligned}$$

(9)

where $\varphi \ge 0$ and $\varphi \in C_0^{\infty }(\Omega _T)$ are available. The usual Caccioppoli estimates are valid.

Lemma 9

(Caccioppoli) Let $p > 2$. Assume that $v \ge 1$ is a weak supersolution in $\Omega _T$. Then the estimate

$$\begin{aligned}&\int _{0}^{T}\!\!\int _{\Omega }\left| \nabla \left( \zeta v^{\frac{p-1-\beta }{p}}\right) \right| ^p\,\mathrm {d}x \,\mathrm {d}t+ \frac{1}{|\beta -1|}\underset{0<t<T}{{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\int _{\Omega }v^{1-\beta }\zeta ^p \,\mathrm {d}x\nonumber \\&\quad \le C(p)\left( \beta ^{p-1}\! +\! \beta ^{1-p}\right) \left\{ \int _{0}^{T}\!\!\int _{\Omega }v^{p-1-\beta }|\nabla \zeta |^p \,\mathrm {d}x \,\mathrm {d}t +\frac{1}{|\beta -1|}\int _{0}^{T}\!\!\int _{\Omega }v^{1-\beta }\Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr | \,\mathrm {d}x \,\mathrm {d}t \right\} \end{aligned}$$

holds for all $\beta > 0$.⁵ For $\beta = 1$ we have

$$\begin{aligned}&\Bigl (\frac{p}{p-2}\Bigr )^p \int _{0}^{T}\!\!\int _{\Omega }\left| \zeta \nabla v^{\frac{p-2}{p}}\right| ^p\,\mathrm {d}x + \underset{0<t<T}{{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\int _{\Omega }\zeta ^p\log (v) \,\mathrm {d}x\\&\quad \le C(p) \int _{0}^{T}\!\!\int _{\Omega }v^{p-2}|\nabla \zeta |^p \,\mathrm {d}x \,\mathrm {d}t +\int _{0}^{T}\!\!\int _{\Omega }\log (v)\Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr | \,\mathrm {d}x \,\mathrm {d}t. \end{aligned}$$

Here $\zeta \ge 0$ is an arbitrary test function in $C_0^{\infty }(\Omega _T)$.

Proof

A formal calculation with the test function $\phi = v^{-\beta }\zeta ^p$ yields the inequality (The cases $\beta \,>\,1$ and $\beta \,<\,1$ are different). See [4, 9, 14]. A variant of Harnack’s inequality is expedient in our present work. It is valid for supersolutions. $\square$

Lemma 10

(Harnack) Let $p>2$. If $v > 0$ is a lower semicontinuos weak supersolution in $B(x_0,4R)\times (0,T),$ then the inequality

(10)

where

is valid at a.e. time $t,\,0<t<T$. Here $c_1 = c_1(n,p),\,c_2 = c_2(n,p)$.

This is Theorem 1.1 in [14]. Note that the waiting time $\tau$ depends on t. The estimate is valid for the so-called Lebesgue times, as explained in [14]. We only need to know that they are dense in (0, T). The convenient notation

is used for the average value.

3 Examples and comments

We shall illustrate the theory with several examples. We begin with a simple observation.

Extension to the past If v is a p-superparabolic function in $\Omega \times (0,T)$ and if $v \ge 0$ there, then the extended function

$$\begin{aligned} v(x,t) = {\left\{ \begin{array}{ll} v(x,t), &{}\text {when} \quad 0<t<T\\ 0,&{} \text {when}\quad t\le 0 \end{array}\right. } \end{aligned}$$

(11)

is p-superparabolic in $\Omega \times (-\infty ,T)$. We use the same notation for the extended function.

The stationary case If $v(x,t)=u(x),$ i.e., v is independent of t,  the equation becomes the elliptic p-Laplace equation

$$\begin{aligned} \nabla \!\cdot \!\bigl (|\nabla u(x)|^{p-2}\nabla u(x)\bigr )\;=\;0 \end{aligned}$$

(12)

in the domain $\Omega$. The p-superparabolic functions become the p-superharmonic functions, defined in [16]. A typical unbounded one is the fundamental solution

$$\begin{aligned} u(x)\;=\;C_{n,p}\left| x-x_0\right| ^{\frac{p-n}{p-1}}, \end{aligned}$$

if $1<p<n$ (The function is bounded if $p>n,$ and the singularity at $x=x_0$ escapes the definition, because it is not an infinity. A singularity it is.)

The infinities can be dense in the domain. We give the example

$$\begin{aligned} v(x,t)\;=\;u(x)\;=\; \sum _{j=1}^{\infty }\,\frac{C_{j,p}}{|x-q_j|^{\frac{n-p}{p-1}}}\quad (2<p<n) \end{aligned}$$

where $q_1,q_2,q_3,\ldots$ are the rational points and the $C_{j,p}$’s are positive convergence factors. The function is, indeed, p-superharmonic in $\mathbb {R}^n$, see [18]. At each rational point

$$\begin{aligned} u(q) \;=\;\lim _{x \rightarrow q}u(x) \;=\;\infty . \end{aligned}$$

This means that $v=v(x,t)$ is a p-superparabolic function taking the value $\infty$ along each rational line $(q,t),\;-\infty < t < \infty$. In this case⁶

$$\begin{aligned} \mathbb {Q}^n\times (-\infty ,\infty ) \;\subset \;\Xi ^{\downarrow }\;=\;\Xi ^{\perp }. \end{aligned}$$

Nonetheless, v is of class $\mathfrak {B}$. In particular, $\nabla v \in L_{loc}^q(\mathbb {R}^n\times \mathbb {R})$ whenever $q < p-1+\tfrac{1}{n+1}$. Now, as always in the stationary case, the exponent has the better range $q < \tfrac{n(p-1)}{n-1}$ according to [16].

The Barenblatt solution This function was treated in the Introduction.

A Separable minorant If $\Omega$ is a bounded regular domain, there exists a p-superparabolic function of the form

$$\begin{aligned} V(x,t) = {\left\{ \begin{array}{ll} \frac{\mathfrak {U}(x)}{(t-t_0)^{\frac{1}{p-2}}}, &{} \text {when}\quad t > t_0\\ 0, &{} \text {when}\quad t \le t_0 \end{array}\right. } \end{aligned}$$

(13)

where $\mathfrak {U} \in C(\overline{\Omega })\cap W^{1,p}(\Omega )$ is a weak solution to the equation

$$\begin{aligned} \nabla \!\cdot \!\bigl (|\nabla \mathfrak {U}|^{p-2}\nabla \mathfrak {U}\bigr )\,+\;\tfrac{1}{p-2}\,\mathfrak {U}\,=\;0 \end{aligned}$$

(14)

and $\mathfrak {U}>0$ in $\Omega$. Moreover, one can take $\mathfrak {U}|_{\partial \Omega } = 0$. The function V is p-parabolic, when $t > t_0$. The solution $\mathfrak {U}$ is unique.⁷ To construct $\mathfrak {U}$, we first minimize the Rayleigh quotient

$$\begin{aligned} R(w)\;=\;\frac{\int _{\Omega }|\nabla w|^p\,\mathrm {d}x}{\Bigl (\int _{\Omega }\!|w|^2\,\mathrm {d}x\Bigr )^{\frac{p}{2}}} \end{aligned}$$

among all functions w in $W^{1,p}_0(\Omega ),\;w\not \equiv 0$. Since $R(|w|) = R(w),$ we may assume that $w\ge 0$. By Sobolev’s and Hölder’s inequalities, $R(w) \ge C(n,p,|\Omega |) > 0$ for all admissible w. The direct method in the Calculus of Variations yields the existence of a minimizer $u \ge 0,\, u \not \equiv 0,$ which satisfies the Euler–Lagrange equation

$$\begin{aligned} \nabla \!\cdot \!\bigl (|\nabla u|^{p-2}\nabla u\bigr )\;+\;\lambda \Vert u\Vert _{L^2(\Omega )}^{p-2}u\;=\;0, \end{aligned}$$

where $\lambda > 0$ is the minimum sought for. We need a normalization. Fix u so that $\Vert u\Vert _{L^2(\Omega )} =1$ and note that Cu satisfies the equation

$$\begin{aligned} \nabla \!\cdot \!\bigl (|\nabla (C u)|^{p-2}\nabla ( Cu)\bigr )\;+\;\lambda \,C^{p-2}(Cu)\;=\;0. \end{aligned}$$

Then choose the constant C so that $\lambda C^{p-2} = \tfrac{1}{p-2}$. The so obtained $\mathfrak {U} = Cu$ is the desired solution. By elliptic regularity theory $\mathfrak {U} \in C(\overline{\Omega })$ and $\mathfrak {U}|_{\partial \Omega } =0$. Finally, since  $\nabla \!\cdot \!\bigl (|\nabla \mathfrak {U}|^{p-2}\nabla \mathfrak {U}\bigr )\;\le 0$   and $\mathfrak {U}\ge 0$ in $\Omega$, Harnack’s inequality for supersolutions of the elliptic p-Laplace equation implies that $\mathfrak {U} > 0$ in $\Omega$. See [20]. We could also prescribe other non-negative boundary values for $\mathfrak {U}$, but these are less needed. In only one space dimension, a formula for $\mathfrak {U}$ is easily obtained.

The constructed function $V = V(x,t)$ is a p-parabolic function, when $t>t_0$. This is a useful property, since it can serve as a minorant. The functions of class $\mathfrak {M}$ have to blow up at least as fast as $(t-t_0)^{-1/(p-2)}$.

Lemma 11

If $v \ge 0$ is a p-superparabolic function in $\Omega _T$ and if

$$\begin{aligned} \lim _{\begin{array}{c} (y,t) \rightarrow (x,0)\\ t>0 \end{array}}v(y,t)\,=\;\infty \end{aligned}$$

for all $x \in \Omega ,$ then

$$\begin{aligned} v(x,t)\, \ge \, \frac{\mathfrak {U}(x)}{t^{\frac{1}{p-2}}}\quad \text {in} \quad \Omega _T. \end{aligned}$$

In particular,

$$\begin{aligned} \liminf _{\begin{array}{c} (y,t) \rightarrow (x,0)\\ t>0 \end{array}}\left( t^{\frac{1}{p-2}}v(y,t)\right) \; > 0\quad \text {in} \quad \Omega . \end{aligned}$$

Proof

The comparison principle yields that

$$\begin{aligned} v(x,t)\, \ge \, \frac{\mathfrak {U}(x)}{(t+\sigma )^{\frac{1}{p-2}}},\quad \text {in}\quad \Omega _{T-\sigma }, \end{aligned}$$

where $\sigma > 0$ is arbitrarily small. Let $\sigma \rightarrow 0$. $\square$

A superposition of a finite number of these functions is possible. Indeed,

$$\begin{aligned} v(x,t) \;=\;\mathfrak {U}(x)\,\sum _{j=1}^{N}\Bigl [\frac{1}{t-t_j}\Bigr ]_+^{\frac{1}{p-2}} \end{aligned}$$

is p-superparabolic (This construction does not work for $N=\infty$.)

The previous Lemma gives the slowest possible growth for p-superparabolic functions of class $\mathfrak {M}$ . But there is no upper bound. The growth can be arbitrarily fast. We just give the example

$$\begin{aligned} v(x,t) = {\left\{ \begin{array}{ll} \mathfrak {U}(x)\exp \Bigl (\frac{1}{(p-2)(t-t_0)}\Bigr ) &{} \text {when}\quad t > t_0\\ 0 &{} \text {when}\quad t \le t_0. \end{array}\right. } \end{aligned}$$

(15)

Here $\Xi ^{\downarrow } = \Omega \times \{t_0\}$. One can even build a tower of exponentials to increase the terrible speed of growth.

Hyperplanes in $\Xi ^{\perp }$ As we have seen, $\Xi ^{\perp }$ and $\Xi ^{\downarrow }$ can contain portions of planes of the form $t = t_0,\,$ so-called time slices. But, surprisingly enough, no planes like

$$\begin{aligned} t = \langle a,x\rangle + t_0, \quad a \not = 0, \end{aligned}$$

will do. Indeed, the associated “supersolution” would be identically $\infty ,$ when $\inf \langle a,x\rangle < t-t_0 < \sup \langle a,x\rangle$. This is outside the realm of p-superparabolic functions, violating the requirement of a dense subset of finite values.

Proof of Proposition 4:

To simplify the exposition, we first treat the case with only one space variable $(n=1)$. Assume that v is p-superparabolic in $(0,2)\times (-\infty ,\infty )$ and that $\Xi ^{\perp }$ contains the line segment

$$\begin{aligned} t= ax,\quad a>0,\quad 0 < x < 2. \end{aligned}$$

This will lead to the contradiction that $v=\infty$ in too large a set. To see this, fix $0 < x_0 < 1,\, t_0 = ax_0$. Let $k >>1$. We claim that

$$\begin{aligned} v(x,t)\;\ge \; k\,\frac{x-x_0}{(t-t_0+\sigma )^{\frac{1}{p-2}}}, \quad \sigma > 0, \end{aligned}$$

in the triangular domain

$$\begin{aligned} x_0 < x < 1,\quad t > ax,\quad t < a\cdot 1. \end{aligned}$$

The claim follows from the comparison principle, because the minorant is a smooth subsolution and the inequality is obviously valid on the parabolic boundary: $x=x_0,\,t \ge t_0;\; x=1,\,t=a;\; t=ax,\, x_0 \le x \le 1$. Send k to $\infty$. As a result, $v = \infty$ in the whole triangular subdomain. This is a contradiction.⁸ This was the case $n=1$.

The proof in several dimensions is rather similar. The equation is invariant under rotations and reflexions of the x-coordinates. Therefore, we may assume that $a_1>0, a_2>0,\cdots ,a_n>0$ in the equation

$$\begin{aligned} t=a_1x_1+a_2x_2+ \cdots +a_nx_n+t_0 \end{aligned}$$

for the plane. The function

$$\begin{aligned} u(x,t) = kt^{-\frac{1}{p-2}}x_1x_2\cdots x_n \qquad \qquad (k>0) \end{aligned}$$

is a p-subparabolic function when $t>0$ and $x_1x_2\ldots x_n >0.$ This is easy to verify by direct calculation, since the function is smooth. On the parabolic boundary of the polyhedral domain

$$\begin{aligned}&0 < a_1x_1+a_2x_2+ \cdots +a_nx_n < t-t_0 <1,\nonumber \\&x_1 > 0, x_2 > 0,\cdots ,x_n > 0 \end{aligned}$$

we have

$$\begin{aligned} v(x,t) \ge k(t-t_0+\sigma )^{-\frac{1}{p-2}}x_1x_2\cdots x_n,\qquad \quad \sigma >0 \end{aligned}$$

for the given p-superparabolic function v,  which we tacitly assume to be defined here. (The boundary consists of parts of $n\!+\!1$ planes, but the plane $t=t_0+1$ is excluded.) By the comparison principle, the inequality holds in the whole polyhedral domain. Letting $k \rightarrow \infty$ we see that $v=\infty$ in an open set, which means that v cannot be finite in a dense subset. This contradiction concludes our proof. $\square$

Fast growth It is easy to exhibit p-superparabolic functions of the form

$$\begin{aligned} v(x,t) = {\left\{ \begin{array}{ll} \mathfrak {U}(x)\Psi (t), &{}\quad t_0 < t < T\\ 0,&{}\quad t \le t_0, \end{array}\right. } \end{aligned}$$

where $\mathfrak {U}$ was constructed in connection with Eq. (14). One example with $T=\infty$ was formula (15).

Solutions that blow up The evolutionary p-Laplace equation has solutions that blow up at a certain time. The example

$$\begin{aligned} \mathfrak {D}(x,t)\;=\;\left\{ A\Bigl (\frac{T}{T-t}\Bigr )^{\frac{n(p-2)}{\lambda (p-1)}}\;+\;\Bigl (\frac{p-2}{p}\Bigr )\lambda ^{-\frac{1}{p-1}}\Bigl (\frac{|x|^p}{T-t}\Bigr )^{\frac{1}{p-1}}\right\} ^{\frac{p-1}{p-2}}, \end{aligned}$$

with $\lambda =n(p-2)+p$, is given in Remark 7.1 on page 331 in [4]. It is a p-parabolic function in $\mathbb {R}^n\times (0,T).$ It blows up at the terminal point $t=T$. As it stays, it is outside the domain, but we can extend $\mathfrak {D}$ into the future, using, for example, the solution (13). Thus

$$\begin{aligned} v(x,t) = {\left\{ \begin{array}{ll} \mathfrak {D}(x,t), &{} \text {when} \quad t < T\\ (t-T)^{-\frac{1}{p-2}}\mathfrak {U}(x),&{} \text {when} \quad t \ge T \end{array}\right. } \end{aligned}$$

is a p-superparabolic function in $\Omega \times (0,\infty ),$ where $\Omega$ comes from the definition of $\mathfrak {U}$. In this case $\Xi ^{\perp } = \Omega \times \{T\}$.

4 Smoothing effects

In this section we shall consider the summability (integrability) of p-superparabolic functions and their gradients. To be more precise, we show that if $v \in L^{p-2+\varepsilon }_{loc}(\Omega _T)$ then v is actually in class $\mathfrak {B}$. We give alternative proofs to those in [10]. The basic tools are the Caccioppoli inequality in Lemma 9 and Sobolev’s inequality, written for convenience in the form

$$\begin{aligned} \int _0^T\!\!\int _{\Omega }\!|\zeta w|^q\,\mathrm {d}x \,\mathrm {d}t\;\le \;S^q \int _0^T\!\!\int _{\Omega }\!|\nabla (\zeta w)|^p\,\mathrm {d}x \,\mathrm {d}t\, \biggl \{ \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{0<t<T}\int _{\Omega }\!|\zeta w|^m\,\mathrm {d}x\biggr \}^{\frac{p}{n}}, \end{aligned}$$

(16)

valid for $m>0$ and $q = p+\tfrac{pm}{n}$. Here $\zeta \in C_0^{\infty }(\Omega _T)$ is a suitable test function and

$$\begin{aligned} \zeta w \in L^{\infty }(0,T;L^m(\Omega )) \cap L^p(0,T;W^{1,p}(\Omega )). \end{aligned}$$

See [4], Proposition 3.1, Chapter 1, page 7. Since our results are local, we may as well assume that the p-superparabolic function v is $\ge 1$ in the whole $\Omega _T$.

The estimates will be obtained by iteration. At each step of the iteration, a new test function $\zeta$ has to be chosen. Typically, the domain shrinks during the procedure. Fortunately, we need only a finite number of steps. Therefore, we do not keep track of the $\zeta$’s. We begin with an alternative proof of a theorem from [10].

Theorem 12

Let v be a p-superparabolic function in $\Omega _T$. If $v \in L^{p-2+\varepsilon }_{loc}(\Omega _T)$ for some $\varepsilon > 0,$ then $v \in L^{p-1+\frac{p}{n}- \sigma }_{loc}(\Omega _T)$ for each $\sigma >0$.

Proof

Since v is superparabolic, it is bounded from below and thus by adding a constant, we may assume that $v\ge 1$. Fix the desired small $\sigma > 0$. Anticipating the procedure, we try to find an index j so that

$$\begin{aligned} \varepsilon \Bigl (1+ \frac{p}{n} \Bigr )^j\;=\; 1 - \frac{\sigma }{1+\frac{p}{n}}. \end{aligned}$$

Since the assumption holds for any $\varepsilon$ smaller than the given one, we can always accomplish this. Let $\xi \in C_0^{\infty }(\Omega _T),\; 0 \le \zeta \le 1,$ and set $\zeta = 1$ in any chosen compact subdomain of $\Omega _T$. The Caccioppoli estimate

$$\begin{aligned}&\int _0^T\!\!\int _{\Omega }\!\left| \nabla \left( \zeta v^{\frac{p-2+\varepsilon }{p}}\right) \right| ^p\,\mathrm {d}x \,\mathrm {d}t\, +\frac{1}{\varepsilon } \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{0<t<T}\int _{\Omega }\!v(x,t)^{\varepsilon }\zeta (x,t)^p\,\mathrm {d}x\\&\quad \le \; \frac{C(p)}{(1-\varepsilon )^p}\biggl \{\int _0^T\!\!\int _{\Omega }\!v^{p-2+\varepsilon }|\nabla \zeta |^p\,\mathrm {d}x \,\mathrm {d}t\;\,+\,\frac{1}{\varepsilon } \int _0^T\!\!\int _{\Omega }\!v^{\varepsilon }\Bigl |\frac{\partial }{\partial t}\zeta ^p \Bigr |\,\mathrm {d}x \,\mathrm {d}t\biggr \} \end{aligned}$$

in Lemma 9 with $\varepsilon = 1-\beta$ is valid when $0 < \varepsilon < 1$. By our assumption, the right-hand side is finite. Combining this with the Sobolev inequality (put $w = v^{\frac{p-2+\varepsilon }{p}}, m=p\varepsilon /(p-2+\varepsilon )$)

$$\begin{aligned}&\int _0^T \int _{\Omega } \zeta ^{p\gamma } v^{p-2+\varepsilon \gamma } \,\mathrm {d}x \,\mathrm {d}t \\&\quad \le \; S^{p\gamma }\int _0^T \int _{\Omega } \left| \nabla \left( \zeta v^{\frac{p-2+\varepsilon }{p}}\right) \right| ^p\,\mathrm {d}x \,\mathrm {d}t \left\{ \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{0<t<T}\int _{\Omega }\!v(x,t)^{\varepsilon }\zeta (x,t)^p\,\mathrm {d}x\right\} ^{\frac{p}{n}} \end{aligned}$$

where $\gamma = 1 + \tfrac{p}{n} \,>\,1,$ we obtain

$$\begin{aligned} \int _0^T\!\!\int _{\Omega }\!\zeta ^{p\gamma } v^{p-2+\varepsilon \gamma }\,\mathrm {d}x \,\mathrm {d}t\;<\;\infty . \end{aligned}$$

Thus

$$\begin{aligned} v \in L_{loc}^{p-2+\varepsilon (1+\frac{p}{n})}(\Omega _T). \end{aligned}$$

We repeat the procedure, now with $\varepsilon (1+\frac{p}{n})$ in the place of $\varepsilon$, and obtain the better exponent

$$\begin{aligned} p-2+\varepsilon \left( 1+\frac{p}{n}\right) \gamma \,=\,p-2+\varepsilon \left( 1+\frac{p}{n}\right) ^2. \end{aligned}$$

Iterating till we reach the exponent $p-2+\varepsilon (1+\tfrac{p}{n})^j,$ we can perform one final iteration, obtaining the desired exponent

$$\begin{aligned} p-2+\varepsilon \left( 1+\frac{p}{n}\right) ^j\gamma \,=\,p-2+\left( 1-\frac{\sigma }{\gamma }\right) \gamma \,=\, p-1+\frac{p}{n}-\sigma . \end{aligned}$$

This concludes our proof, but we remark that an explicit estimate can be worked out, which we omit, since only a finite number of iterations was needed. $\square$

In the next theorem from [10], it is decisive that one can deduce that $\nabla v \in L_{loc}^{p-1}(\Omega _T),$ because this is sufficient to induce a Radon measure. For the benefit of the reader, we give a proof.

Theorem 13

Let $v \in L_{loc}^{p-2+\varepsilon }(\Omega _T)$ be a p-superparabolic function in $\Omega _T$. Then the Sobolev gradient $\nabla v$ exists and $\nabla v \in L_{loc}^{q}(\Omega _T)$ whenever $q< p-1+\tfrac{1}{n+1}$.

Proof

The proof is the same as in [16]. By [9, 17] or [12] the gradient exists. Let $0<t_1<t_2<T$ and $K\subset \subset \Omega$. Take $0<\beta <1$. By the Hölder inequality

$$\begin{aligned}&\int _{t_1}^{t_2}\!\!\int _K\!|\nabla v|^q\,\mathrm {d}x \,\mathrm {d}t\\&\quad =\; \int _{t_1}^{t_2}\!\!\int _K\!\left( v^{-\frac{1+\beta }{p}} |\nabla v|\right) ^qv^{\frac{1+\beta }{p}q}\,\mathrm {d}x \,\mathrm {d}t\\&\quad \le \; \biggl \{\int _{t_1}^{t_2}\!\!\int _K\!v^{-1-\beta }|\nabla v|^p\,\mathrm {d}x \,\mathrm {d}t \biggr \}^{\frac{q}{p}}\biggl \{\int _{t_1}^{t_2}\!\!\int _K\!v^{\frac{1+\beta }{p-q}q}\,\mathrm {d}x \,\mathrm {d}t\biggr \}^{1-\frac{q}{p}}\\&\quad =\;\Bigl (\frac{p}{p-1-\beta }\Bigr )^q\biggl \{\int _{t_1}^{t_2}\!\!\int _K\!|\nabla \left( v^{\frac{p-1-\beta }{p}}\right) |^p\,\mathrm {d}x \,\mathrm {d}t\biggr \}^{\frac{q}{p}}\biggl \{\int _{t_1}^{t_2}\!\!\int _K\!v^{\frac{1+\beta }{p-q}q}\,\mathrm {d}x \,\mathrm {d}t \biggr \}^{1-\frac{q}{p}}. \end{aligned}$$

The last integral is finite if

$$\begin{aligned} (1+\beta )\frac{q}{p-q}\,<\;p-1+\frac{p}{n} \end{aligned}$$

by the previous theorem, and the first one by the Caccioppoli estimate, whenever $0 < \beta < 1$. We see that any exponent

$$\begin{aligned} q < p-1+\frac{1}{n+1} \end{aligned}$$

is possible to reach. $\square$

Remark

Also the opposite implication holds: if $\nabla v \in L_{loc}^q(\Omega _T)$ when $q < p-1+\tfrac{1}{n+1},$ then $v \in L_{loc}^{p-2+\varepsilon }(\Omega _T),$ when $1 + \frac{p}{n} >\varepsilon > 0$.

For the Barenblatt solution (2) the integrals

$$\begin{aligned} \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{-\infty <t<\infty }\int _{\mathbb {R}^n}\!\mathfrak {B}(x,t)^{\alpha }\,\mathrm {d}x \end{aligned}$$

converge when $0 <\alpha \le 1$ but not when $\alpha > 1$. We have the following general result, characterizing Class $\mathfrak {B}$.

Theorem 14

Suppose that v is p-superparabolic in $\Omega _T$. If $v \ge 1$ and

$$\begin{aligned} \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{0<t<T}\int _{\Omega }\!v(x,t)^{\alpha }\,\mathrm {d}x \, <\; \infty \end{aligned}$$

for some exponent $\alpha >0,$ then $v \in L^s_{loc}(\Omega _T)$ whenever $s<p-1+\tfrac{p}{n}$.

Remark

As a consequence,

$$\begin{aligned} \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{0<t<T}\int _{\Omega }\!v(x,t)^{\alpha }\,\mathrm {d}x \, =\; \infty \end{aligned}$$

for all exponents $\alpha >0,$ if v belongs to $\mathfrak {M}$.

Proof

We shall show that $v \in L^s_{loc}(\Omega _T)$ for some $s \ge p-2$. If $s > p-2$ we are done, because Theorem 12 now applies. To be on the safe side, we first treat the case $s=p-2$. Then $v \in L^{p-2}_{loc}(\Omega _T)$ and the Caccioppoli inequality

$$\begin{aligned}&\int _{0}^{T}\!\!\int _{\Omega }\left| \nabla \left( \zeta v^{\frac{p-2}{p}}\right) \right| ^p\,\mathrm {d}x \,\mathrm {d}t \\&\quad \le C_1(p)\int _{0}^{T}\!\!\int _{\Omega }\Bigl (v^{p-2}|\nabla \zeta |^p + \log (v)\Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr |\bigr ) \,\mathrm {d}x \,\mathrm {d}t\\&\quad \le C_2(p)\int _{0}^{T}\!\!\int _{\Omega }v^{p-2}\Bigl (|\nabla \zeta |^p + \Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr |\bigr ) \,\mathrm {d}x \,\mathrm {d}t < \infty \end{aligned}$$

in Lemma 9 is available. In the Sobolev inequality (16) we take $w = v^{\frac{p-2}{p}}$ and $m = \tfrac{\alpha p}{p-2},$ so that $w^m = v^{\alpha }$ in the single integral. Then

$$\begin{aligned} q= p\Bigl (1+\frac{p\alpha }{n(p-2)}\Bigr ), \qquad w^q = v^{p-2+\frac{\alpha p}{p-2}}. \end{aligned}$$

It follows that $v \in L_{loc}^{p-2+\frac{p \alpha }{n}}(\Omega _T)$ and now the summability exponent is in the range for which Theorem 12 is applicable. That much about the case $s=p-2$.

Next we describe an iteration, starting the procedure from some small $\alpha$ in the range $0 < \alpha < p-2$. The Caccioppoli inequality

$$\begin{aligned}&\int _{0}^{T}\!\!\int _{\Omega }\left| \nabla \left( \zeta v^{\frac{\alpha }{p}}\right) \right| ^p\,\mathrm {d}x \,\mathrm {d}t \\&\quad \le \, \;C_3(p)\frac{(p-1-\alpha )^{p-1}}{p-2-\alpha }\int _{0}^{T}\!\!\int _{\Omega }v^{\alpha -(p-2)}\Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr |\,\mathrm {d}x \,\mathrm {d}t\\&\qquad + C_3(p) \int _{0}^{T}\!\!\int _{\Omega } v^{\alpha }|\nabla \zeta |^p \,\mathrm {d}x \,\mathrm {d}t \\&\quad \le \,\frac{C_4(p)}{p-2-\alpha }\int _{0}^{T}\!\!\int _{\Omega }v^{\alpha }\Bigl (|\nabla \zeta |^p + \Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr |\Bigr ) \,\mathrm {d}x \,\mathrm {d}t \; < \; \infty \end{aligned}$$

is at our disposal (We used $v^{\alpha -(p-2)} \le v^{\alpha }$ in the last step). With $w=v^{\frac{\alpha }{p}}$ and $m=p$, we can use the Sobolev inequality. Now

$$\begin{aligned} q= p\Bigl (1+\frac{p}{n}\Bigr ), \quad w^q = v^{\alpha (1+ \frac{p}{n})}. \end{aligned}$$

It follows that $v \in L_{loc}^{\alpha (1+\frac{p}{n})}(\Omega _T)$. If $\alpha (1+\tfrac{p}{n}) < p-2$ we repeat the procedure, this time with

$$\begin{aligned} w = v^{\alpha \frac{\left( 1+\frac{p}{n}\right) }{p}},\quad m = \frac{p}{1+\frac{p}{n}},\quad q = p\Bigl [1 + \frac{p}{n(1+\frac{p}{n})}\Bigr ] \end{aligned}$$

so that

$$\begin{aligned} w^q = v^{\alpha \left( 1+\frac{2p}{n}\right) }. \end{aligned}$$

Thus the result is that $v \in L_{loc}^{\alpha (1+\frac{2p}{n})}\!(\Omega _T)$. We continue till we, sooner or later, reach an index j for which

$$\begin{aligned} \alpha \Bigl (1+\frac{jp}{n}\Bigr )\;<p-2 \quad \text {and} \quad \alpha \Bigl (1+\frac{(j+ 1)p}{n}\Bigr )\; \ge p-2. \end{aligned}$$

We can do one final iteration using $w = v^{\alpha (1+\frac{jp}{n})/p},\;m= \tfrac{p}{1+\tfrac{jp}{n}}$. It follows that $v \in L_{loc}^s(\Omega _T)$ for some $s \ge p-2$. This case was dealt with above. $\square$

5 Class $\mathfrak {M}$

A typical p-superparabolic function which is not of Class $\mathfrak {B}$ is the previously constructed

$$\begin{aligned} V(x,t)\,=\,\left[ \frac{1}{t-t_0}\right] _+^{\frac{1}{p-2}}\mathfrak {U}(x) \end{aligned}$$

in $\Omega \times (-\infty ,\infty ),$ where $\Omega$ has to be bounded. This function is not locally summable to any power $\ge p-2,$ nor is its gradient. For a given function v,  defined in $\Omega _T$, we recall that

$$\begin{aligned} \Xi ^{\perp }(t_0)\;=\;&\bigl \{x\in \Omega |\; \lim _{\begin{array}{c} (y,t) \rightarrow (x,t_0)\\ t>t_0 \end{array}}v(y,t)\;=\;+\infty \bigr \}\\ \Xi ^{\downarrow }(t_0)\;=\;&\bigl \{x\in \Omega |\; \lim _{t \rightarrow t_0+}v(x,t)\;=\;+\infty \bigr \}, \end{aligned}$$

and so

$$\begin{aligned} \Xi ^{\perp }\;=\;\bigcup _{0\le t<T}\Xi (t),\quad \Xi ^{\downarrow }\,=\; \bigcup _{0\le t<T}\Xi ^{\downarrow }(t). \end{aligned}$$

Of course, $\Xi ^{\perp }(t_0) \subset \Xi ^{\downarrow }(t_0),$ but they do not have to be the same sets, as the Barenblatt solution shows. The striking phenomenon is that if the n-dimensional measure $|\Xi ^{\downarrow }(t_0)|\,>\,0,$ then also $|\Xi ^{\perp }(t_0)|\,>\,0$. Before dealing with this, we need to give the following lemma about large average values.

Lemma 15

Suppose that v is a non-negative p-superparabolic function in $\Omega _T$. Suppose that $B(x_0,4R) \subset \Omega$. If there is a sequence of “Lebesgue times” $t_j \rightarrow t_0, \;0<t_j<T$ such that

$$\begin{aligned} \lim _{j \rightarrow \infty } \int _{B(x_0,R)}\!v(x,t_j)\,\mathrm {d}x \quad =\quad \infty , \end{aligned}$$

then

$$\begin{aligned} v(x,t)\;\ge \; \gamma \, \frac{R^{\frac{p}{p-2}}}{(t-t_0)^{\frac{1}{p-2}}}, \end{aligned}$$

when $x \in B(x_0,2R)$ and $t_0 < t < T$. The constant $\gamma > 0$ depends only on n and p.

Remark

If $t_0>0$ we do not forbid that $t_j < t_0$.

Proof

We aim at using Harnack’s inequality (10) for the bounded supersolutions $v_k,$ where k does not have to be an integer. Now, for a fixed index j, by continuity the integral

attains all values in the interval when k increases from 0 to $\infty$. Let $S>t_0$ be a given number so that $S-t_0$ is small enough. Then, for j large enough,

by our assumption. Determine $k_j$ (not necessarily an integer) so that

$$\begin{aligned} J^{k_j}(t_j)\;=\;\left( \frac{c_1R^p}{S-t_0}\right) ^{\frac{1}{p-2}}. \end{aligned}$$

By Harnack’s inequality (10) evaluated at the Lebesgue time $t_j$ we have

$$\begin{aligned}&\left( \frac{c_1R^p}{S-t_0}\right) ^{\frac{1}{p-2}}\;\le \; \frac{1}{2}\left( \frac{c_1R^p}{S-t_j}\right) ^{\frac{1}{p-2}}\;+\; c_2\,\inf _{Q^j_{2R}}v,\\&\text {where}\\&Q^j_{2R}\;=\; B(x_0,2R) \times \bigl (t_j+\frac{\tau _j}{2},t_j+\tau _j\bigr ),\\&\tau _j \;=\;\min \bigl \{S-t_j,c_1R^pJ^{k_j}(t_j)^{2-p}\bigr \}\;=\;\min \{S-t_j,S-t_0\}. \end{aligned}$$

Taking the limit as $j \rightarrow \infty ,$ we arrive at

$$\begin{aligned} \frac{1}{2}\left( \frac{c_1R^p}{S-t_0}\right) ^{\frac{1}{p-2}}\;\le \;c_2\,\inf _{Q_{2R}} v, \end{aligned}$$

where

$$\begin{aligned} Q_{2R}\;=\; B(x_0,2R) \times \Bigl (\frac{t_0+S}{2},S\Bigr ). \end{aligned}$$

Then we have the inequality

$$\begin{aligned} c_2\, v(x,t)\;\ge \;\frac{1}{2}\left( \frac{c_1R^p}{S-t_0}\right) ^{\frac{1}{p-2}}\;\ge \; \frac{1}{2}\left( \frac{c_1R^p}{2(t-t_0)}\right) ^{\frac{1}{p-2}}, \end{aligned}$$

when $S > t >\tfrac{S+t_0}{2}$. By adjusting S we can reach all t in $(t_0,T)$. $\square$

Corollary 16

If, at some point $(x_0,t_0),$

$$\begin{aligned} \liminf _{\begin{array}{c} (x,t)\rightarrow (x_0,t_0)\\ t>t_0 \end{array}}\bigl ((t-t_0)^{\frac{1}{p-2}}v(x,t)\bigr )\;>\;0, \end{aligned}$$

then  $\Xi ^{\perp }(t_0)\,=\,\Omega \times \{t_0\}$.

Proof

In some small neighborhood $|x-x_0| \le \rho ,\,t_0 < t <t_0 + \rho ^p,$ we have

$$\begin{aligned} (t-t_0)^{\frac{1}{p-2}}v(x,t)\;\ge \;\varepsilon _0\;>\;0. \end{aligned}$$

Then

as $t \rightarrow t_0+$. The assumption in Lemma 15 is fulfilled. The inclusion $B(x_0,2\rho )\;\subset \;\Xi ^{\perp }(t_0)$ follows. We can apply Lemma 15 on any ball $B(y_0,R)$ with $B(y_0,4R) \subset \subset \Omega$ intersecting $B(x_0,2\rho )$ and conclude that also $B(y_0,R)\; \subset \;\Xi ^{\perp }(t_0)$. Using a suitable chain of balls, we can see that the corollary holds. $\square$

If there are too much infinities inside the domain, they have to touch the lateral boundary: the infinities “cast a shadow.” That is in the next theorem.

Theorem 17

If for some $t_0$ the n-dimensional measure $|\Xi ^{\downarrow }(t_0)|\,>\,0,$ then also $|\Xi ^{\perp }(t_0)|\,>\,0$. Actually,

$$\begin{aligned} \Xi ^{\perp }(t_0)\;=\;\Xi ^{\downarrow }(t_0)\;=\; \Omega \times \{t_0\}. \end{aligned}$$

Proof

We take $t_0=0$ and select some ball $B(x_0,8R)$ in $\Omega$ so that

$$\begin{aligned} \left| \Xi ^{\downarrow }(0)\cap B(x_0,R)\right| \,>\,0, \end{aligned}$$

which is possible by the assumption. Let $k > 0$. To each $x \in \Xi ^{\downarrow }(t_0)$ there is a time $t^k_x>0$ such that

$$\begin{aligned} v(x,t) > k,\quad \text {when} \quad 0 < t < t^k_x. \end{aligned}$$

We remark that the times $t^k_x$ are decreasing as $k \rightarrow \infty$. Define the line segments

$$\begin{aligned} L^k_x\;=\; \{x\}\times (0,t^k_x) \end{aligned}$$

and consider the projected sets

$$\begin{aligned} E^k_t\;=\; \overline{B(x_0,R)}\cap \bigl \{x|\;(x,t)\in \cup _y L_y^k\bigr \}, \end{aligned}$$

which consists of all endpoints $x \in \overline{B(x_0,R)}$ with the corresponding $t^k_x > t$. The set $E^k_t$ shrinks with increasing k.

Claim: There is a $T^k > 0$ such that

$$\begin{aligned} |E^k_t|\;\ge \; \frac{1}{2}\,|\Xi ^{\downarrow }\cap B(x_0,R)| \quad \text {when}\quad 0 < t < T^k. \end{aligned}$$

Moreover, $T^k$ decreases when $k \rightarrow \infty$.

Indeed,

$$\begin{aligned} \Xi ^{\downarrow }(0) \cap B(x_0,R)\,=\; \bigcap _{k=1}^{\infty }\bigcup _{j=1}^{\infty }E_{\frac{1}{j}}^k \end{aligned}$$

so that

$$\begin{aligned} \left| \Xi ^{\downarrow }(0) \cap B(x_0,R)\right| \;\le \;\left| \bigcup _{j=1}^{\infty }E_{\frac{1}{j}}^k\right| \;=\; \lim _{j \rightarrow \infty }\left| E_{\frac{1}{j}}^k\right| , \end{aligned}$$

since the sets are nested. To each k there is a $j=j_k$ such that

$$\begin{aligned} \left| E_{\frac{1}{j_k}}^k\right| \,>\; \frac{1}{2}\,\left| \Xi ^{\downarrow }(0) \cap B(x_0,R)\right| . \end{aligned}$$

The claim follows, because

$$\begin{aligned} E_{\frac{1}{j_k}}^k \subset E^k_t,\quad \text {when}\quad 0 < t < \tfrac{1}{j_k} = T^k. \end{aligned}$$

The so defined $T^k$ are decreasing, if we select the $j_k$ to be increasing.

We select a compact subset $\tilde{E}^t_k \subset E^t_k$ so that $|\tilde{E}^t_k|\,\ge \, \tfrac{1}{2}|E^t_k|$. Thus

$$\begin{aligned} \left| \tilde{E}^t_k\right| \,\ge \, \frac{1}{4}\,\left| \Xi ^{\downarrow }(0) \cap B(x_0,R)\right| \quad \text {when}\quad 0 < t< T^k. \end{aligned}$$

Let $0 < t< T^k$ and $x \in \tilde{E}^t_k$. By the semicontinuity, there is a radius $r^k_{x,t} < \frac{R}{10}$ such that

$$\begin{aligned} \inf _{Q^k_{x,t}}v\;\ge \; k,\qquad Q^k_{x,t}\;=\;B\left( x,r^k_{x,t}\right) \times \left[ t,t+\left( r^k_{x,t}\right) ^p\right) . \end{aligned}$$

Obviously,

$$\begin{aligned} \tilde{E}^t_k\;\subset \bigcup _{x\in \tilde{E}^t_k}B\left( x,r^k_{x,t}\right) . \end{aligned}$$

By compactness of $\tilde{E}^t_k$ and by a simple version of Vitali’s covering theorem ([19], Chapter I, paragraph 1.6) there is finite $J_k$ and disjoint balls $B(x_j,r_j^k)$ so that

$$\begin{aligned} \tilde{E}^t_k\;\subset \bigcup _{j=1}^{J_k}B\left( x_j,5r^k_j\right) . \end{aligned}$$

If $t< \tau < t + [\min \{r_1^k,r^k_2,\ldots r^k_{J_k}\}]^p$ then

$$\begin{aligned}&\int _{B(x_0,2R)}\!\min \{v(x,\tau ),k\}\,\mathrm {d}x\;\ge \; \int _{\bigcup B(x_j,r^k_j)}\min \{v(x,\tau ),k\}\,\mathrm {d}x\\&\quad =\;\sum _{j=1}^{J_k}\int _{B(x_j,r^k_j)}\min \{v(x,\tau ),k\}\,\mathrm {d}x\;\ge \; k\,\sum _{j=1}^{J_k}\left| B\left( x_j,r^k_j\right) \right| \\&\quad =\;k 5^{-n}\sum _{j=1}^{J_k}\left| B\left( x_j,5r^k_j\right) \right| \;\ge \; k5^{-n}|\tilde{E}^t_k|\;\ge \; k4^{-1}5^{-n}|\Xi ^{\downarrow }(0)\cap B(x_0,R)|. \end{aligned}$$

Thus

$$\begin{aligned} \int _{B(x_0,2R)}\!\min \{v(x,t),k\}\,\mathrm {d}x\;\ge \;\frac{k}{4\cdot 5^n}\,\left| \Xi ^{\downarrow }(0) \cap B(x_0,R)\right| \end{aligned}$$

for $0 < t < T^k$. From Lemma 15 it follows that

$$\begin{aligned} \lim _{\begin{array}{c} (y,t)\rightarrow (x,0)\\ t>0 \end{array}}v(y,t)\;=\;\infty \quad \text {for all}\quad x \in B(x_0,4R). \end{aligned}$$

In other words,

$$\begin{aligned} B(x_0,4R)\;\subset \;\Xi ^{\perp }(0)\;\subset \;\Xi ^{\downarrow }(0). \end{aligned}$$

We read off from the proof that the infinities in $B(x_0,R)$ cause that the whole larger ball $B(x_0,4R)$ consists of infinities at time $t=0$. Repeating the argument with suitable chains of balls, we can conclude that the whole $\Omega \times \{0\}$ consists of infinities (We have assumed that $\Omega$ is connected). $\square$

We saw that $|\Xi ^{\perp }(t)| \,=\,0$ and $|\Xi ^{\downarrow }(t)|\,=\,0$ simultaneously. Positive measure led to the situation with the violent behavior described in Sect. 5. Yet, to complete the picture, we need to show that, if $|\Xi ^{\perp }(t)| \,=\,0$ for each $0<t<T,$ then the function belongs to Class $\mathfrak {B}$. By Theorem 14 it is enough to establish the following.

Lemma 18

If $|\Xi ^{\perp }(t)| \,=\,0$ for each $0<t<T,$ then $v \in L^{\infty }_{loc}(0,T;L_{loc}^1(\Omega ))$.

Proof

The antithesis is that

$$\begin{aligned} \mathop {{{\mathrm{\mathop {{ess\,sup}}\limits }}}}\limits _{\varepsilon <t<T-\varepsilon }\int _{B(x_0,R)}\!v(x,t)\,\mathrm {d}x\,=\; \infty \end{aligned}$$

for some R and $\varepsilon$. We can extract a convergent sequence of Lebesgue times, say $t_j \rightarrow t_0,$ such that

$$\begin{aligned} \lim _{j \rightarrow \infty }\int _{B(x_0,R)}\!v(x,t_j)\,\mathrm {d}x\,=\; \infty . \end{aligned}$$

Lemma 15 implies that

$$\begin{aligned} \lim _{\begin{array}{c} (y,t)\rightarrow (x,t_0)\\ t > t_0 \end{array}}v(y,t) \;=\;\infty \quad \text {for all}\quad x \in B(x_0,2R). \end{aligned}$$

Thus $B(x_0,2R)\,\subset \, \Xi ^{\perp }(t_0)$ and so $|\Xi ^{\perp }(t_0)|\,>0$. This contradiction shows that the antithesis is false. The lemma follows. $\square$

If $\Omega \times (0,T)=\mathbb {R}^n\times (0,T)$, then $\Xi ^\perp (0)$ is of measure zero.

Theorem 19

If $v:\mathbb {R}^n\times (0,T) \rightarrow [0,\infty ]$ is p-superparabolic, then the n-dimensional measure $|\Xi ^{\perp }(0)|\,=\,0$.

Proof

Assume that $|\Xi ^{\perp }(0)|\,>\,0$. We can regard v as zero, when $t \le 0$. There must be a point where Corollary 16 applies, thus $\Xi ^{\perp }(0)\,=\,\mathbb {R}^n\times \{0\}$. Choose an arbitrarily large ball B(0, R) and let

$$\begin{aligned} \mathfrak {V}(x,t)\;=\; t^{-\frac{1}{p-2}}\mathfrak {U}(x) \end{aligned}$$

be the p-parabolic function constructed in the unit ball $|x| < 1,$ as in formula (13). By scaling and comparison

$$\begin{aligned} v(x,t)\;\ge \;\Bigl (\frac{R^p}{t}\Bigr )^{\frac{1}{p-2}}\mathfrak {U}\bigl (\frac{x}{R}\bigr ),\quad t >0,\; |x| < R. \end{aligned}$$

Let $\nu \,=\,\min _{|y|\le 1/2}\mathfrak {U}(y)\; >\; 0$. Then

$$\begin{aligned} v(x,t)\;\ge \; \nu \Bigl (\frac{R^p}{t}\Bigr )^{\frac{1}{p-2}} \quad \text {when} \quad |x| \le \frac{R}{2}. \end{aligned}$$

Letting $R \rightarrow \infty ,$ we must have $v(x,t) \equiv \infty$. The function is not finite in a dense subset. $\square$

6 The “Outsiders”

The p-superparabolic functions do not form a closed class under monotone convergence. In the stationary case, the limit of an increasing sequence of p-superharmonic functions is either identically infinite or a p-superharmonic function. For the Evolutionary p-Laplace equation, the situation is not quite that simple. The limit of an increasing sequence of p-superparabolic functions can be a function that is identically infinite in some time intervals:

$$\begin{aligned} v(x,t)\, \equiv \infty \quad \text {when} \quad x \in \Omega ,\; t_1 < t < t_2. \end{aligned}$$

This follows from our previous considerations, because the truncations

$$\begin{aligned} \min \{v(x,t),k\} \end{aligned}$$

are bounded p-superparabolic functions. It is also possible to construct examples such that

$$\begin{aligned} \Xi ^{\perp }\; \supseteq \;\Omega \times \bigcup _{j} [a_j,b_j], \end{aligned}$$

where the union of disjoint time intervals is countable. We can use estimate (5) to conclude that

$$\begin{aligned} \iint _{\Omega _T\cap \{v<\infty \}}\!|\nabla \log v|^p\zeta ^p \,\mathrm {d}x \,\mathrm {d}t\; \le&\; c\int _0^T\!\!\int _{\Omega }v^{2-p}\Bigl |\frac{\partial \zeta ^p}{\partial t}\Bigr |\,\mathrm {d}x \,\mathrm {d}t\, \\&\; + c \int _0^T\!\!\int _{\Omega }|\nabla \zeta |^p \,\mathrm {d}x \,\mathrm {d}t \end{aligned}$$

for strictly positive v (If $v \equiv \infty$, there is nothing to say).