Communications on Pure and Applied Analysis (CPAA)

On the critical exponents for porous medium equation with a localized reaction in high dimensions

Pages: 649 - 658, Volume 11, Issue 2, March 2012      doi:10.3934/cpaa.2012.11.649

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Zhilei Liang - School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China (email)

Abstract: This paper is concerned with the critical exponents for the porous medium equation

$u_{t}=\triangle u^m+a(x)u^p, (x,t)\in R^N\times (0,T), $

where $m>1, p>0,$ and the function $a(x)\geq 0$ has a compact support. Suppose the space dimension $N\geq 2$, we prove that the global exponent $p_0$ and the Fujita type exponent $p_c$ are both $m$: if $0 < p < m$ every solution is global in time, if $ p = m $ all the solutions blow up and if $p > m$ both the blowing up solutions and the global solutions exist. While for the one-dimensional case, it is proved $p_0=\frac{m+1}{2} < m+1 = p_c$ by [E. Ferreira, A. Pablo, J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231(2006) 195-211].

Keywords:  Porous media, critical exponents, blow up, localized reaction.
Mathematics Subject Classification:  Primary: 35K15, 35K55; Secondary: 35K65.

Received: September 2010;      Revised: June 2011;      Available Online: October 2011.