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

Sachiko Ishida

doi:10.3934/proc.2013.2013.335

Article Contents

2013, Volume 2013: 335-344. Doi: 10.3934/proc.2013.2013.335 open access

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

Sachiko Ishida^1,

1.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: July 31, 2012

Revised: October 31, 2012

Published: October 2013

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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.

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References

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