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

 

Global existence and nonexistence for degenerate parabolic systems


Authors: Yuxiang Li, Weibing Deng and Chunhong Xie
Journal: Proc. Amer. Math. Soc. 130 (2002), 3661-3670
MSC (2000): Primary 35K50, 35K55, 35K65
Published electronically: May 14, 2002
MathSciNet review: 1920046
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Abstract: The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system

\begin{displaymath}\begin{split} u_t=v^p(\Delta u+au),\\ v_t=u^q(\Delta v+bv) \end{split}\end{displaymath}

in the cylinder $\Omega\times(0,\infty)$, where $\Omega\subset R^N$ is bounded and $p, q, a, b$ are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by $\lambda_1$ the first Dirichlet eigenvalue for the Laplacian on $\Omega$. We prove that there exists a global solution iff $\lambda_1\geq \min\{a,b\}$.


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Additional Information

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

Weibing Deng
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06630-3
PII: S 0002-9939(02)06630-3
Keywords: Global existence, global nonexistence, degenerate parabolic system
Received by editor(s): July 23, 2001
Published electronically: May 14, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society