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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Weibing Deng, Yuxiang Li and Chunhong Xie
Journal: Proc. Amer. Math. Soc. 131 (2003), 1573-1582
MSC (2000): Primary 35K50, 35K55, 35K65
Published electronically: 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 $p, q, a, b$ 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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Additional Information

Weibing Deng
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: wbdeng@nju.edu.cn

Yuxiang Li
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: lieyuxiang@yahoo.com.cn

Chunhong Xie
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06866-1
PII: S 0002-9939(02)06866-1
Keywords: Global existence-nonexistence, degenerate parabolic system, non-local
Received by editor(s): January 8, 2002
Published electronically: December 16, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society