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Archive for Rational Mechanics and Analysis

, Volume 194, Issue 2, pp 383–419 | Cite as

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

  • Naoyuki Ichihara
  • Hitoshi IshiiEmail author
Article

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.

Keywords

Viscosity Solution Asymptotic Solution Jacobi Equation Representation Formula Pointwise Convergence 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Graduate School of Natural EngineeringHiroshima UniversityHiroshimaJapan
  2. 2.Department of Mathematics, Faculty of Education and Integrated Arts and SciencesWaseda UniversityTokyoJapan

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