Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations

TL;DR

This paper studies the long-term behavior of viscosity solutions for certain coupled Hamilton--Jacobi systems on a torus, proving convergence to asymptotic solutions under specific conditions.

Contribution

It provides new convergence results for quasi-monotone weakly coupled Hamilton--Jacobi systems, extending understanding of their large-time dynamics.

Findings

Proved convergence of viscosity solutions to asymptotic solutions.

Established results under restricted assumptions.

Analyzed systems on the n-dimensional torus.

Abstract

We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the $n$-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.