Differential and Integral Equations

Classification of solutions of porous medium equation with localized reaction in higher space dimensions

Xiaosong Kang, Wenbiao Wang, and Xiaofang Zhou

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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.

Article information

Differential Integral Equations, Volume 24, Number 9/10 (2011), 909-922.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations 35B44: Blow-up


Kang, Xiaosong; Wang, Wenbiao; Zhou, Xiaofang. Classification of solutions of porous medium equation with localized reaction in higher space dimensions. Differential Integral Equations 24 (2011), no. 9/10, 909--922. https://projecteuclid.org/euclid.die/1356012892

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