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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.

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Authors and Affiliations

  1. Graduate School of Natural Engineering, Hiroshima University, Hiroshima, Japan

    Naoyuki Ichihara

  2. Department of Mathematics, Faculty of Education and Integrated Arts and Sciences, Waseda University, Tokyo, Japan

    Hitoshi Ishii

Authors
  1. Naoyuki Ichihara
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  2. Hitoshi Ishii
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Correspondence to Hitoshi Ishii.

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Communicated by P.-L. Lions

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Ichihara, N., Ishii, H. Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians. Arch Rational Mech Anal 194, 383–419 (2009). https://doi.org/10.1007/s00205-008-0170-0

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  • DOI: https://doi.org/10.1007/s00205-008-0170-0

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