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.
Yuzhu Han, Wenjie Gao, and Haixia Li
Jilin University, Jilin University, Jilin University
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
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