Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton–Jacobi equations

Abstract

Consider the diffusive Hamilton–Jacobi equation

$$\begin{aligned} u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text { in } \Omega \times (0,T) \end{aligned}$$

with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with $p>2$. We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question by establishing the optimal estimate

$$\begin{aligned} C_1(T-t)^{-1/(p-2)}\le \Vert \nabla u(t)\Vert _\infty \le C_2(T-t)^{-1/(p-2)} \end{aligned}$$

(1)

for time-increasing gradient blowup solutions in any convex, smooth bounded domain $\Omega$ with $2<p<3$. We also cover the case of (nonradial) solutions in a ball for $p=3$. Moreover we obtain the almost sharp rate in general (nonconvex) domains for $2<p\le 3$. The proofs rely on suitable auxiliary functionals, combined with the following, new Bernstein-type gradient estimate with sharp constant:

$$\begin{aligned} |\nabla u|\le d_\Omega ^{-1/(p-1)}\bigl (d_p+C d_\Omega ^\alpha \bigr ) \ \ \text { in } \Omega \times (0,T),\qquad d_p=(p-1)^{-1/(p-1)}, \end{aligned}$$

(2)

where $d_\Omega$ is the function distance to the boundary. This close connection between the temporal and spatial estimates (1) and (2) seems to be a completely new observation. Next, for any $p>2$, we show that more singular rates may occur for solutions which are not time-increasing. Namely, for a suitable class of solutions in one space-dimension, we prove the lower estimate $\Vert u_x(t)\Vert _\infty \ge C(T-t)^{-2/(p-2)}$.