Localization of blow-up points for a nonlinear nonlocal porous medium equation

Lili Du; Zheng-An Yao

doi:10.3934/cpaa.2007.6.183

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2007, Volume 6, Issue 1: 183-190. Doi: 10.3934/cpaa.2007.6.183

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Localization of blow-up points for a nonlinear nonlocal porous medium equation

Lili Du^1, and
Zheng-An Yao¹

1.
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275

Received: November 30, 2005

Revised: July 31, 2006

Published: February 2007

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Abstract

This paper deals with the porous medium equation with a nonlinear nonlocal source

$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$

subject to homogeneous Dirichlet condition. We investigate the influence of the nonlocal source and local term on blow-up properties for this system. It is proved that: (i) when $p\leq 1$, the nonlocal source plays a dominating role, i.e. the system has global blow-up and the blow-up profile which is uniformly away from the boundary either at polynomial scale or logarithmic scale is obtained. (ii) When $p > m$, this system presents single blow-up pattern. In other words, the local term dominates the nonlocal term in the blow-up profile. This extends the work of Li and Xie in Appl. Math. Letter, 16 (2003) 185--192.

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Mathematics Subject Classification: 35B40, 35K65.

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