## Differential and Integral Equations

### Asymptotic solutions with slow convergence rate of Hamilton-Jacobi equations in Euclidean $n$ space

#### Abstract

In this paper, we study the long-time asymptotics of the Cauchy problem for the Hamilton-Jacobi equation $$u_t(x,t) + \alpha x \cdot Du(x,t) + H(Du(x,t)) = f(x) \ \ \text{ in } \ \ {\mathbb R }^n \times (0,\infty) ,$$ where $\alpha$ is a positive constant. In \cite{FIL2}, it was shown that there are a constant $c \in {\mathbb R }$ and a viscosity solution $v$ of $c + \alpha x\cdot Dv(x) + H(Dv(x)) = f(x)$ in ${\mathbb R }^n$ such that $u(\cdot,t) -(v(\cdot)+ct) \to 0$ as $t \to \infty$ locally uniformly in ${\mathbb R }^n$. The function $v(x) + ct$ is called the asymptotic solution. Our goal is to give a sufficient condition in order that the set of points where the rate of this convergence is slower than $t^{-1}$ is non-empty. We also give several examples which show that we can not remove, in general, the assumptions in this sufficient condition in order that this set is non-empty. As a result, we clarify crucial factors which cause this slow rate of convergence. They are both a geometrical property of the set of equilibrium points and a lower bound of the initial data.

#### Article information

Source
Differential Integral Equations Volume 20, Number 10 (2007), 1185-1200.

Dates
First available in Project Euclid: 20 December 2012

Fujita, Yasuhiro; Uchiyama, Kazuya. Asymptotic solutions with slow convergence rate of Hamilton-Jacobi equations in Euclidean $n$ space. Differential Integral Equations 20 (2007), no. 10, 1185--1200.https://projecteuclid.org/euclid.die/1356039302