Asymptotic profiles of solutions to viscous Hamilton–Jacobi equations

Abstract

The large time behavior of solutions to the Cauchy problem for the viscous Hamilton–Jacobi equation u t −Δ u+|∇ u| q =0 is classified. If q> q c :=( N+2)/( N+1), it is shown that non-negative solutions corresponding to integrable initial data converge in W 1,p( R N) as t→∞ toward a multiple of the fundamental solution for the heat equation for every p∈[1,∞] (diffusion-dominated case). On the other hand, if 1< q< q c , the large time asymptotics is given by the very singular self-similar solution of the viscous Hamilton–Jacobi equation. For non-positive and integrable solutions, the large time behavior of solutions is more complex. The case q⩾2 corresponds to the diffusion-dominated case. The diffusion profiles in the large time asymptotics appear also for q c < q<2 provided suitable smallness assumptions are imposed on the initial data. Here, however, the most important result asserts that under some conditions on initial data and for 1< q<2, the large time behavior of solutions is given by the self-similar viscosity solutions to the non-viscous Hamilton–Jacobi equation z t +|∇ z| q =0 supplemented with the initial datum z( x,0)=0 if x≠0 and z(0,0)<0.