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

$L^\infty$-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, 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.
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, Basel, 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), 1421-1440.

[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), 2469-2491.

[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), 367-400.

[6]

E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.

[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), 597-621.

[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), 1279-1297.

[9]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations 20 (2007), 133-180.

[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), 333-364.

[11]

R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection, Hokkaido Mathematical Journal 27 (1998), 147-196.

show all references

References:
[1]

H. Amann, "Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory'', Birkhäuser, Basel, 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), 1421-1440.

[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), 2469-2491.

[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), 367-400.

[6]

E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.

[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), 597-621.

[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), 1279-1297.

[9]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations 20 (2007), 133-180.

[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), 333-364.

[11]

R. Suzuki, Existence and nonexistence of global solutions to quasilinear parabolic equations with convection, Hokkaido Mathematical Journal 27 (1998), 147-196.

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