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

Sachiko Ishida; Tomomi Yokota

doi:10.3934/proc.2013.2013.345

Article Contents

2013, Volume 2013: 345-354. Doi: 10.3934/proc.2013.2013.345 open access

This issue Previous Article $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems Next Article Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization

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

Sachiko Ishida^1, and
Tomomi Yokota^1,

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

Received: August 2012

Revised: December 2012

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

The global existence of weak solutions to quasilinear ``degenerate'' Keller-Segel systems is shown in the recent papers [3], [4]. This paper gives some improvements and supplements of these. More precisely, the differentiability and the smallness of initial data are weakened when the spatial dimension $N$ satisfies $N\geq2$. Moreover, the global existence is established in the case $N=1$ which is unsolved in [4].

Keywords:

Quasilinear degenerate Keller-Segel systems,
global existence.

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B33.

Citation:

Related Papers

Cited by

References

[1]	T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183-217.
[2]	S. Ishida, A study on the solvability of degenerate Keller-Segel systems, Ph.D. thesis.
[3]	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.
[4]	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.
[5]	E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399-415.
[6]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal. 10 (1986), 299-314.
[7]	Y. Sugiyama, Global existence in the sub-critical cases and finite time blow-up in the super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations 19 (2006), 841-876.
[8]	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.
[9]	Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 252 (2012), 692-715.
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010), 2889-2905.