A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians

TL;DR

This paper introduces a novel approach using the nonlinear adjoint method to analyze the large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, including parabolic and coupled systems.

Contribution

It develops a new machinery for studying large time behavior, applicable to a broad class of Hamilton--Jacobi equations, with robust and adaptable methods.

Findings

Established convergence results for large time behavior

Identified new long time averaging effects

Demonstrated robustness of the methods

Abstract

We introduce a new machinery to study the large time behavior for general classes of Hamilton--Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.