Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(1): 55-66

Printed January 1, 2014

https://doi.org/10.4134/BKMS.2014.51.1.55

Copyright © The Korean Mathematical Society.

Extinction and non-extinction of solutions to a fast diffusive $p$-Laplace equation with a nonlocal source

Yuzhu Han, Wenjie Gao, and Haixia Li

Jilin University, Jilin University, Jilin University

Abstract

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive $p$-Laplace equation $u_t=\mathrm{div}(|\nabla u|^{p-2}\nabla u)+a\int_\Omega u^q(y,t) dy $, $1 < p < 2$, in a bounded domain $\Omega\subset R^N$ with $N \geq 1$. More precisely, it is shown that if $ q > p - 1 $, any solution vanishes in finite time when the initial datum or the coefficient $a$ or the Lebesgue measure of the domain is small, and if $0 < q < p-1$, there exists a solution which is positive in $\Omega$ for all $t>0$. For the critical case $q=p-1$, whether the solutions vanish in finite time or not depends crucially on the value of $a\mu$, where $\mu=\int_{\Omega}\phi^{p-1}(x)\mathrm{d}x$ and $\phi$ is the unique positive solution of the elliptic problem $-\mathrm{div}(|\nabla \phi|^{p-2}\nabla \phi)=1$, $x\in \Omega$; $\phi(x)=0$, $x\in\partial\Omega$. This is a main difference between equations with local and nonlocal sources.

Keywords: $p$-Laplace equation, nonlocal source, extinction

MSC numbers: 35B33, 35K57