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

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

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
Related Papers

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. 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), 252* (2012), 1421. 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), 252* (2012), 2469. Google Scholar
[5]	E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol. 26 (1970), 26 (1970), 399. Google Scholar
[6]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal. 10 (1986), 10 (1986), 299. 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), 19 (2006), 841. 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), 227* (2006), 333. Google Scholar
[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. Google Scholar
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations 248* (2010), 248* (2010), 2889. Google Scholar

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. 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), 252* (2012), 1421. 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), 252* (2012), 2469. Google Scholar
[5]	E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol. 26 (1970), 26 (1970), 399. Google Scholar
[6]	M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations,, Nonlinear Anal. 10 (1986), 10 (1986), 299. 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), 19 (2006), 841. 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), 227* (2006), 333. Google Scholar
[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. Google Scholar
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations 248* (2010), 248* (2010), 2889. Google Scholar

[1]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[2]	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
[3]	Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537
[4]	Sachiko Ishida. An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems. Conference Publications, 2015, 2015 (special) : 635-643. doi: 10.3934/proc.2015.0635
[5]	Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
[6]	Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
[7]	Chao Deng, Tong Li. Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 183-195. doi: 10.3934/dcdsb.2018093
[8]	Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[9]	Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181
[10]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[11]	Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038
[12]	Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
[13]	Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464
[14]	Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[15]	Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 211-232. doi: 10.3934/dcdss.2020012
[16]	Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[17]	Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015
[18]	Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891
[19]	Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117
[20]	Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 119-137. doi: 10.3934/dcdss.2020007

Impact Factor:

American Institute of Mathematical Sciences