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

 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

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].
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), 183-217.  Google Scholar [2] S. Ishida, A study on the solvability of degenerate Keller-Segel systems,, Ph.D. thesis., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [5] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399-415. Google Scholar [6] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations, Nonlinear Anal. 10 (1986), 299-314.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 252 (2012), 692-715.  Google Scholar [10] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010), 2889-2905.  Google Scholar

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