Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

Olivier Ley (1, 2) , Vinh Duc Nguyen (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes
3 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12

We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some ``asymmetric systems'' where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems.

Mots clés: Nonlinear degenerate parabolic equations; Nonlinear degenerate elliptic equations; Hamilton-Jacobi equations; monotone systems; gradient bounds; oscillation; strong maximum principle; ergodic problem; asymptotic behavior; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Ley, Vinh Duc Nguyen. Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems. Nonlinear Analysis: Theory, Methods and Applications, 2016, 130, pp.76-101. ⟨10.1016/j.na.2015.09.012⟩. ⟨hal-01022962v2⟩