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Blow-up phenomena for a class of fourth-order parabolic problems

Author: G. A. Philippin
Journal: Proc. Amer. Math. Soc. 143 (2015), 2507-2513
MSC (2010): Primary 35K30, 35K57
Published electronically: February 16, 2015
MathSciNet review: 3326032
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Abstract: This paper deals with some qualitative properties of solutions to a class of semilinear fourth-order parabolic problems. It is shown that under certain conditions on the data, $ u({\bf x}, t)$ cannot exist for all time and an upper bound for $ t^{\star }$ is derived where $ (0,t^{\star })$ is the interval of existence of $ u({\bf x}, t)$. Moreover we construct (under certain conditions on the data) a lower bound for $ t^{\star }$ when blow-up occurs. This last result is based on some Sobolev type inequality established at the end of the paper.

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Additional Information

G. A. Philippin
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec, Canada G1V OA6

Keywords: Fourth-order parabolic problems, Blow-up, Sobolev type inequality
Received by editor(s): November 20, 2013
Received by editor(s) in revised form: January 12, 2014
Published electronically: February 16, 2015
Communicated by: Michael Hitrik
Article copyright: © Copyright 2015 American Mathematical Society