Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions

Abstract

Global classical solutions to the viscous Hamilton–Jacobi equation u_t−Δu=a|∇u|^p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W^1,∞(Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.