American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
  • DCDS-B Home
  • This Issue
  • Next Article
    Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model
December  2013, 18(10): 2537-2568. doi: 10.3934/dcdsb.2013.18.2537

Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$

Sachiko Ishida 1, , Yusuke Maeda 1, and  Tomomi Yokota 1,

1. 

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

Received  November 2012 Revised  February 2013 Published  October 2013

This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.
Keywords: blow-up., Quasilinear non degenerate Keller-Segel systems.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B3.
Citation: 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
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar

[2]

H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (1995).  doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations,, Comm. Partial Differential Equations, 22 (1997), 1647.  doi: 10.1080/03605309708821314.  Google Scholar

[5]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[6]

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

[7]

H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319.  doi: 10.1007/s002080050224.  Google Scholar

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1968).   Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[10]

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.   Google Scholar

[11]

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

[12]

Y. Sugiyama and 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.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[13]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar

[2]

H. Amann, Linear and Quasi-linear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (1995).  doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis., J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations,, Comm. Partial Differential Equations, 22 (1997), 1647.  doi: 10.1080/03605309708821314.  Google Scholar

[5]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[6]

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

[7]

H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains,, Math. Ann., 312 (1998), 319.  doi: 10.1007/s002080050224.  Google Scholar

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, (1968).   Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[10]

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.   Google Scholar

[11]

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

[12]

Y. Sugiyama and 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.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[13]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  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]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
[9]

Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
[10]

Vincent Calvez, Thomas O. Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1175-1208. doi: 10.3934/dcds.2016.36.1175
[11]

Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029
[12]

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
[13]

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
[14]

Jaewook Ahn, Kyungkeun Kang. On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5165-5179. doi: 10.3934/dcds.2014.34.5165
[15]

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
[16]

Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 211-232. doi: 10.3934/dcdss.2020012
[17]

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
[18]

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
[19]

Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[20]

Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences