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

TL;DR

This paper establishes new bounds on oscillation and gradients for viscosity solutions of fully nonlinear degenerate elliptic equations, and applies these results to analyze the large-time behavior of certain coupled parabolic systems.

Contribution

It introduces novel oscillation and gradient bounds for degenerate elliptic equations and extends their application to the asymptotic analysis of asymmetric coupled systems.

Findings

Bounds applicable to systems with mixed sublinear and superlinear Hamiltonians

Analysis of large-time behavior of weakly coupled nonlinear parabolic systems

Extension of maximum principle to degenerate systems

Abstract

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.