Differential and Integral Equations

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

Yasuhiro Fujita and Kazuya Uchiyama

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039302

Mathematical Reviews number (MathSciNet)
MR2365208

Zentralblatt MATH identifier
1212.35208

Subjects
Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 35B40: Asymptotic behavior of solutions 49L25: Viscosity solutions

Citation

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.


Export citation