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

Pages: 183 - 190, Volume 6, Issue 1, March 2007

doi:10.3934/cpaa.2007.6.183       Abstract        Full Text (130.8K)       Related Articles

Lili Du - Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China (email)
Zheng-An Yao - Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China (email)

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.

Keywords:  Porous medium equation, nonlinear nonlocal source, blow-up, blow-up set.
Mathematics Subject Classification:  35B40, 35K65.

Received: December 2005;      Revised: August 2006;      Published: December 2006.