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

 

Blow-up for degenerate parabolic equations with nonlocal source


Authors: Youpeng Chen, Qilin Liu and Chunhong Xie
Journal: Proc. Amer. Math. Soc. 132 (2004), 135-145
MSC (2000): Primary 35K55, 35K57, 35K65
Published electronically: May 9, 2003
MathSciNet review: 2021256
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Abstract: This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma}u_{x})_{x}=\int_{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain.


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Additional Information

Youpeng Chen
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Address at time of publication: Department of Mathematics, Yancheng Teachers College, Yancheng 224002, Jiangsu, People’s Republic of China
Email: youpengchen@263.sina.net, youpchen@yahoo.com.cn

Qilin Liu
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/S0002-9939-03-07090-4
PII: S 0002-9939(03)07090-4
Keywords: Degenerate nonlocal problem, classical solution, global existence, blow-up set
Received by editor(s): August 20, 2002
Published electronically: May 9, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society