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2005, 2005(Special): 443-452. doi: 10.3934/proc.2005.2005.443

Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent

Michinori Ishiwata 1,

1. 

Department of Mathematical Sciences,, School of Science and Engineering, Waseda University, 169-855, 4-1 Okubo 3-chome, Shinjyuku-ku, Tokyo, Japan

Received  September 2004 Revised  April 2005 Published  September 2005

In this paper, we discuss the asymptotic behavior of some solutions for nonlinear parabolic equation in ${\Bbb R}^N$ involving critical Sobolev exponent. For the subcritical problem (with bounded domain), it is well-known that the solution which intersects the "stable set" must be a global one. But for the critical problem, it is not known whether the same conclusion holds or not. In this paper, we shall show that, in the critical case, the same conclusion actually holds true. The proof requires the concentration compactness type argument.
Keywords: lack of compactness, Stable set, critical Sobolev exponent, concentration-compactness principle..
Mathematics Subject Classification: Primary: 35B33, 35B35; Secondary: 35K6.
Citation: Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443
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