# American Institute of Mathematical Sciences

October  2017, 37(10): 5151-5162. doi: 10.3934/dcds.2017223

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

 Suzhou University of Science and Technology, Suzhou 215009, China

* Corresponding author: Xia Li

Received  September 2016 Revised  April 2017 Published  June 2017

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

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 $H(x, u(x), Du(x))= 0$.

Citation: Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223
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