#### Vol. 298, No. 1, 2019

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The asymptotic bounds of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations

### Kaizhi Wang

Vol. 298 (2019), No. 1, 217–232
DOI: 10.2140/pjm.2019.298.217
##### Abstract

We study the Cauchy problem for time-periodic Hamilton–Jacobi equations with Tonelli Hamiltonians. It is well known that the Cauchy problem admits a unique bounded viscosity solution. We provide a more precise description of the boundedness of the viscosity solution. We introduce the notion of asymptotic bounds of the viscosity solution of the Cauchy problem. An asymptotic bound is a 1-periodic viscosity solution of the Hamilton–Jacobi equation. We show how to obtain the optimal asymptotic bounds, i.e., minimal asymptotic upper bound and maximal asymptotic lower bound. Our method relies upon Mather theory and weak KAM theory on Lagrangian dynamics.

##### Keywords
Hamilton–Jacobi equations, viscosity solutions, Cauchy problem, asymptotic bounds, weak KAM theory
##### Mathematical Subject Classification 2010
Primary: 35B40, 35F25, 37J99