Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Weibing Deng; Yuxiang Li; Chunhong Xie
Journal: Proc. Amer. Math. Soc. 131 (2003), 1573-1582.
MSC (2000): Primary 35K50, 35K55, 35K65
Posted: December 16, 2002
MathSciNet review: 1949888
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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 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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