A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi Equations

TL;DR

This paper studies the long-term behavior of solutions to boundary-value Hamilton-Jacobi equations with nonconvex Hamiltonians, using PDE techniques to establish convergence under broad conditions.

Contribution

It introduces a PDE-based method to analyze large-time asymptotics for various boundary conditions in nonconvex Hamilton-Jacobi equations, extending existing results.

Findings

Viscosity solutions converge to asymptotic solutions as time approaches infinity.

Results hold under general Hamiltonian conditions and weak domain assumptions.

Applicable to Neumann, oblique, and Dirichlet boundary conditions.

Abstract

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.