, Volume 194, Issue 2, pp 383-419
Date: 10 Sep 2008

Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians

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Abstract

We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \({\mathbb{R}^n}\) with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.

Communicated by P.-L. Lions