# American Institute of Mathematical Sciences

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

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.
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). [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. [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. [4] S. Ishida, T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, submitted., (). [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. [6] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol. 26 (1970), 26 (1970), 399. [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. [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. [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. [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. [11] R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection,, Hokkaido Mathematical Journal 27 (1998), 27 (1998), 147.

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##### References:
 [1] H. Amann, "Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory'',, Birkhäuser, (1995). [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. [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. [4] S. Ishida, T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, submitted., (). [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. [6] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol. 26 (1970), 26 (1970), 399. [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. [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. [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. [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. [11] R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection,, Hokkaido Mathematical Journal 27 (1998), 27 (1998), 147.
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