Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions

Xia Li

doi:10.3934/dcds.2017223

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2017, Volume 37, Issue 10: 5151-5162. Doi: 10.3934/dcds.2017223

This issue Previous Article A global bifurcation theorem for a positone multiparameter problem and its application Next Article Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions

Xia Li^,

Suzhou University of Science and Technology, Suzhou 215009, China

* Corresponding author: Xia Li
* Corresponding author: Xia Li

Received: September 2016

Revised: April 2017

Published: October 2017

The first author is supported by National Natural Science Foundation of China (Grant 11471238)

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Abstract

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

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Mathematics Subject Classification: Primary: 35D40, 35F21; Secondary: 49R99.

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