Large time behavior for some nonlinear degenerate parabolic equations

Olivier Ley (1) , Vinh Duc Nguyen (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Asymptotic behavior; Nonlinear degenerate parabolic equations; Hamilton-Jacobi equations; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Ley, Vinh Duc Nguyen. Large time behavior for some nonlinear degenerate parabolic equations. Journal de Mathématiques Pures et Appliquées, 2014, 102 (2), pp.293-314. ⟨10.1016/j.matpur.2013.11.010⟩. ⟨hal-00829824⟩