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

2019/12/02 by Attouchi, Amal, Souplet, Philippe · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.00626

Abstract

Consider the diffusive Hamilton-Jacobi equation u_t-Δu=|∇ u|^p+h(x) in Ω×(0,T) 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 C₁(T-t)^-1/(p-2)≤ ‖∇ u(t)‖_∞ ≤ C₂(T-t)^-1/(p-2) for time-increasing gradient blowup solutions in any convex, smooth bounded domain Ω with 22, 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 ‖u_x(t)‖_∞ ≥ C(T-t)^-2/(p-2).

Cited by

• Mean Field Games of Controls with Boundary Conditions & Invariance Constraints
• Classification of entire and ancient solutions of the diffusive Hamilton-Jacobi equation

Related

• The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation
• A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain
• Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with p-Laplacian diffusion
• Sharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations
• Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion