Global existence and asymptotic behavior for diffusive Hamilton-Jacobi equations with Neumann boundary conditions

TL;DR

This paper proves that solutions to a diffusive Hamilton-Jacobi equation with Neumann boundary conditions exist globally, remain bounded, and exponentially converge to a constant, extending previous results to more general nonlinearities and removing convexity restrictions.

Contribution

It establishes global existence, boundedness, and exponential convergence for solutions of the diffusive Hamilton-Jacobi equation with Neumann conditions, generalizing previous polynomial convergence results and domain convexity assumptions.

Findings

Solutions exist globally and are bounded.

Solutions converge exponentially to a constant.

Results apply to a broad class of nonlinearities.

Abstract

We investigate the diffusive Hamilton-Jacobi equation $$u_t-\Lap u = |\nabla u|^p$$ with $p>1$, in a smooth bounded domain of $\RN$ with homogeneous Neumann boundary conditions and $W^{1,\infty}$ initial data. We show that all solutions exist globally, are bounded and converge in $W^{1,\infty}$ norm to a constant as $t\to\infty$, with a uniform exponential rate of convergence given by the second Neumann eigenvalue. This improves previously known results, which provided only an upper polynomial bound on the rate of convergence and required the convexity of the domain. Furthermore, we extend these results to a rather large class of nonlinearities $F(\nabla u)$ instead of~$|\nabla u|^p$.