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

Pages: 443 - 452, Issue Special, August 2005

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

Abstract: 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:  Stable set, critical Sobolev exponent, lack of compactness, concentration-compactness principle.
Mathematics Subject Classification:  Primary: 35B33, 35B35; Secondary: 35K65.

Received: September 2004;      Revised: April 2005;      Published: September 2005.