Localization of blow-up points for a nonlinear nonlocal porous medium equation
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.
Received: December 2005; Revised: August 2006; Published: December 2006.
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