Existence and nonexistence of global solutions of some non-local degenerate parabolic systems

Deng, Weibing; Li, Yuxiang; Xie, Chunhong

doi:10.1090/S0002-9939-02-06866-1

Existence and nonexistence of global solutions of some non-local degenerate parabolic systems
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by Weibing Deng, Yuxiang Li and Chunhong Xie PDF

Proc. Amer. Math. Soc. 131 (2003), 1573-1582 Request permission

Abstract:

This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system \begin{align*} u_t&=v^p\left (\Delta u+a\int _\Omega v dx\right ), v_t&=u^q\left (\Delta v+b\int _\Omega u dx\right ),\quad x\in \Omega , t>0, \end{align*} with homogeneous Dirichlet boundary conditions, where $\Omega \subset \mathbb {R}^N$ is a bounded domain with a smooth boundary $\partial \Omega$ and $p, q, a, b$ are positive constants. For all initial data, it is proved that there exists a global positive solution iff $\int _\Omega \varphi (x) dx\leq 1/\sqrt {ab}$, where $\varphi (x)$ is the unique positive solution of the linear elliptic problem $-\Delta \varphi (x)=1, x\in \Omega ; \varphi (x)=0, x\in \partial \Omega .$

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