Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

TL;DR

This paper investigates the long-term behavior of solutions to convex Hamilton-Jacobi equations with Neumann boundary conditions, proving uniform convergence to an asymptotic solution over bounded domains.

Contribution

It establishes the uniform convergence of solutions to an asymptotic state for convex Hamilton-Jacobi equations with Neumann boundary conditions.

Findings

Solutions converge uniformly to an asymptotic solution as time approaches infinity.

The Hamiltonian's convexity and coercivity are key to the convergence result.

The results apply to bounded domains with Neumann boundary conditions.

Abstract

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u_t(x,t)+H(x,Du(x,t))=0 in \Omega \times (0,\infty), where \Omega is a bounded open subset of R^n, with Hamiltonian H=H(x,p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t goes to \infty.