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$L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems

Pages: 335 - 344, Issue special, November 2013

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Sachiko Ishida - Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan (email)

Abstract: This paper deals with quasilinear degenerate Keller-Segel systems of parabolic-elliptic type. In this type, Sugiyama-Kunii [10] established the $L^r$-decay ($1\leq r<\infty$) of solutions with small initial data when $q\geq m+\frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity). However, the $L^\infty$-decay property was not obtained yet. This paper gives the $L^\infty$-decay property in the super-critical case with small initial data.

Keywords:  Degenerate parabolic-elliptic Keller-Segel systems, $L^\infty$-decay.
Mathematics Subject Classification:  Primary: 35K57; Secondary: 35B33.

Received: August 2012;      Revised: November 2012;      Published: November 2013.

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