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

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Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems

2013, 2013(special): 335-344. doi: 10.3934/proc.2013.2013.335

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

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

Received August 2012 Revised November 2012 Published November 2013

Abstract
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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: 35B3.

Citation: Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335

References:

[1]	H. Amann, "Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory'',, Birkhäuser, (1995). Google Scholar
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[3]	S. Ishida, T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations 252* (2012), 252* (2012), 2469. Google Scholar
[4]	S. Ishida, T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, submitted., (). Google Scholar
[5]	T. Kawanago, Existence and behavior of solutions for $u_t=\Delta(u^m)+u^l$,, Adv. Math. Sci. Appl. 7 (1997), 7 (1997), 367. Google Scholar
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[10]	Y. Sugiyama, H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations 227* (2006), 227* (2006), 333. Google Scholar
[11]	R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection,, Hokkaido Mathematical Journal 27 (1998), 27 (1998), 147. Google Scholar

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References:

[1]	H. Amann, "Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory'',, Birkhäuser, (1995). Google Scholar
[2]	S. Ishida, T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421. Google Scholar
[3]	S. Ishida, T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations 252* (2012), 252* (2012), 2469. Google Scholar
[4]	S. Ishida, T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, submitted., (). Google Scholar
[5]	T. Kawanago, Existence and behavior of solutions for $u_t=\Delta(u^m)+u^l$,, Adv. Math. Sci. Appl. 7 (1997), 7 (1997), 367. Google Scholar
[6]	E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol. 26 (1970), 26 (1970), 399. Google Scholar
[7]	S. Luckhaus, Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems,, M2AN Math. Model. Numer. Anal. 40 (2006), 40 (2006), 597. Google Scholar
[8]	S. Luckhaus, Y. Sugiyama, Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases,, Indiana Univ. Math. J. 56 (2007), 56 (2007), 1279. Google Scholar
[9]	Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis,, Differential Integral Equations 20 (2007), 20 (2007), 133. Google Scholar
[10]	Y. Sugiyama, H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations 227* (2006), 227* (2006), 333. Google Scholar
[11]	R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection,, Hokkaido Mathematical Journal 27 (1998), 27 (1998), 147. Google Scholar

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