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

 

Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy


Author: Chunpeng Wang
Journal: Proc. Amer. Math. Soc. 141 (2013), 3125-3140
MSC (2010): Primary 35K65, 35D30, 35B33
Published electronically: May 9, 2013
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Abstract: This paper concerns the asymptotic behavior of solutions to a semilinear parabolic equation with boundary degeneracy. It is proved that for the problem in a bounded domain with a homogeneous boundary condition, there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while the nontrivial solution must blow up in a finite time if the degeneracy is strong enough. For the problem in an unbounded domain, blowing-up theorems of Fujita type are established and the critical Fujita exponent is finite in the not strong degeneracy case, while infinite in the other case. Furthermore, the behavior of solutions at the degenerate point is studied, and it is shown that for the nontrivial initial datum vanishing at the degenerate point, the solution always vanishes at the degenerate point if the degeneracy is strong enough, while never if it is not.


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Additional Information

Chunpeng Wang
Affiliation: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: wangcp@jlu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11945-3
PII: S 0002-9939(2013)11945-3
Keywords: Boundary degeneracy, asymptotic behavior
Received by editor(s): May 21, 2011
Received by editor(s) in revised form: November 14, 2011
Published electronically: May 9, 2013
Additional Notes: The author was supported by the NNSF (grants No. 11071100 and 11222106), FANEDD of China and 985 Program of Jilin University.
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.