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

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2013, 2013(special): 345-354. doi: 10.3934/proc.2013.2013.345

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
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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: global existence., Quasilinear degenerate Keller-Segel systems.

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B3.

Citation: Sachiko Ishida, Tomomi Yokota. Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 345-354. doi: 10.3934/proc.2013.2013.345

References:

[1]	T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol. 58 (2009), 58 (2009), 183.
[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), 252* (2012), 1421.
[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), 252* (2012), 2469.
[5]	E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol. 26 (1970), 26 (1970), 399.
[6]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal. 10 (1986), 10 (1986), 299.
[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), 19 (2006), 841.
[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), 227* (2006), 333.
[9]	Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations 252* (2012), 252* (2012), 692.
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations 248* (2010), 248* (2010), 2889.

show all references

References:

[1]	T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol. 58 (2009), 58 (2009), 183.
[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), 252* (2012), 1421.
[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), 252* (2012), 2469.
[5]	E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol. 26 (1970), 26 (1970), 399.
[6]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal. 10 (1986), 10 (1986), 299.
[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), 19 (2006), 841.
[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), 227* (2006), 333.
[9]	Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations 252* (2012), 252* (2012), 692.
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations 248* (2010), 248* (2010), 2889.

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