Mathematics

CLC number: O193

A PDE Approach to the Long-Time Asymptotic Solutions of Contact Hamilton-Jacobi Equations

School of Mathematics Science, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China

^† To whom correspondence should be addressed. E-mail: lixia0527@188.com

Received: 14 October 2021

Abstract

We study the long-time asymptotic behaviour of viscosity solutions $u(x,t)$ of the Hamilton-Jacobi equation $u_t(x,t)+H(x,u(x,t),Du(x,t))=\mathrm{0}$ in ${\mathbb{T}}^n\times (\mathrm{0},\mathrm{\infty })$ with a PDE approach, where $H=H(x,u,p)$ is coercive in $p$, non-decreasing in $u$ and strictly convex in $(u,p)$, and establish the uniform convergence of $u(x,t)$ to an asymptotic solution $u_{\mathrm{\infty }}(x)$ as $t\to \mathrm{\infty }$. Moreover, $u_{\mathrm{\infty }}$ is a viscosity solution of Hamilton-Jacobi equation $H(x,u(x),Du(x))=\mathrm{0}$.