Large time behavior for some nonlinear degenerate parabolic equations

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 Sigma 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 Sigma and, on Sigma, 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 Sigma. As a consequence, we prove the convergence of some everywhere degenerate second-order equations. (C) 2013 Elsevier Masson SAS. All rights reserved.