A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations

TL;DR

This paper presents a novel PDE method for analyzing the long-term behavior of Hamilton-Jacobi solutions, simplifying previous approaches under a refined strict convexity condition on Hamiltonians.

Contribution

It introduces a new PDE approach that generalizes and simplifies existing methods for large time asymptotics of Hamilton-Jacobi equations using refined convexity assumptions.

Findings

The new approach simplifies existing proofs.

It broadens the class of Hamiltonians covered.

It encompasses previous results as special cases.

Abstract

We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations, which modifies and simplifies the previous ones (Barles and Souganidis, 2000; Barles, Ishii and Mitake, 2012), under a refined "strict convexity" assumption on the Hamiltonians. Not only such "strict convexity" conditions generalize the corresponding requirements on the Hamiltonians in Barles and Souganidis (2000), but also one of the most refined our conditions covers the situation studied in Namah and Roquejoffre (1999).