Extinction and non-extinction of solutions to a fast diffusive $p$-Laplace equation with a nonlocal source
Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 55-66
https://doi.org/10.4134/BKMS.2014.51.1.55
Printed January 1, 2014
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
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