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September/October 2011 Classification of solutions of porous medium equation with localized reaction in higher space dimensions
Xiaosong Kang, Wenbiao Wang, Xiaofang Zhou
Differential Integral Equations 24(9/10): 909-922 (September/October 2011).

Abstract

We consider the behavior of nonnegative solutions to the Cauchy problem of the porous medium equation with localized reaction term: \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = \Delta(u^m) + a(x)u^p, & (x,t) \in \mathbf{R}^n \times (0,T),\\ u(x,0) = u_0(x), & x \in \mathbf{R}^n, \end{array} \right. \end{eqnarray*} where $ m > 1 $, $ p > 0 $, $ a(x) \geq 0 $ is a compactly supported function, and $u_0(x)$ is continuous, nonnegative but not identical with zero, and has compact support as well. We show the relationship between the occurrence of blow-ups and the exponents $m$ and $p $: in two-dimensional space, all the solutions are globally defined if $0 < p \leq \frac{m+1}{2}$, and the solutions may blow up in finite time if $p \geq m$; in spaces higher than two-dimensional, all the solutions are global if $0 < p < m$, and there exist both global solutions and blow-up solutions if $p \geq m$. We also show that, for any solution, the intersection of its support and the support of $a(x)$ will be non-empty at some time.

Citation

Download Citation

Xiaosong Kang. Wenbiao Wang. Xiaofang Zhou. "Classification of solutions of porous medium equation with localized reaction in higher space dimensions." Differential Integral Equations 24 (9/10) 909 - 922, September/October 2011.

Information

Published: September/October 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35157
MathSciNet: MR2850345

Subjects:
Primary: 35B44, 35K57

Rights: Copyright © 2011 Khayyam Publishing, Inc.

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Vol.24 • No. 9/10 • September/October 2011
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