Large time behavior for some nonlinear degenerate parabolic equations

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear degenerate parabolic equations, establishing convergence results under various degeneracy and convexity conditions, including nonconvex Hamiltonians.

Contribution

It extends convergence analysis to a broad class of degenerate equations, relaxing uniform parabolicity and handling nonconvex Hamiltonians.

Findings

Proves convergence for equations with degeneracy on set S.

Handles nonconvex Hamiltonians under certain assumptions.

Establishes convergence for fully degenerate second-order equations.

Abstract

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining S as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside S and, on S, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside S. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.