American Institute of Mathematical Sciences

Advanced
March  2015, 14(2): 383-396. doi: 10.3934/cpaa.2015.14.383

Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function

Qi Wang 1,

1. 

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130

Received  November 2013 Revised  September 2014 Published  December 2014

We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not too large.
Keywords: logarithmic sensitivity., Chemotaxis, global existence.
Mathematics Subject Classification: Primary: 92C17, 35B32, 35B35, 35B36, 35B40, 35J4.
Citation: Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure & Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, \emph{Function Spaces, (1993), 9.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, \emph{Advances in Mathematical Sciences and Applications}, 9 (1999), 347.   Google Scholar

[3]

M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis,, \emph{Bioessays}, 28 (2006), 9.   Google Scholar

[4]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, \emph{Math. Bioscience}, 56 (1983), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[5]

D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation,, \emph{Current Opinion in Genetics Development}, 16 (2006), 367.   Google Scholar

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag-Berlin-New York, (1981).   Google Scholar

[7]

D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I,, \emph{Jahresber DMV}, 105 (2003), 103.   Google Scholar

[8]

D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II,, \emph{Jahresber DMV}, 106 (2003), 51.   Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis,, \emph{Journal of Mathematical Biology}, 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius,, \emph{Discrete Contin. Dyn. Syst-Series B}, 7 (2007), 125.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system,, \emph{J. Diff. Equation}, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model,, \emph{Journal of Mathematical Biology}, 35 (1996), 583.  doi: 10.1007/s002850050049.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability,, \emph{Journal of Theoratical Biology}, 26 (1970), 399.   Google Scholar

[14]

E. F. Keller and L. A. Segel, Model for Chemotaxis,, \emph{Journal of Theoratical Biology}, 30 (1971), 225.   Google Scholar

[15]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, \emph{Journal of Theoratical Biology}, 30 (1971), 235.   Google Scholar

[16]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, \emph{SIAM J. Appl. Math.}, 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[17]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, \emph{Journal of Mathematical Biology}, 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[19]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, \emph{Adv. Math. Soc. Appl.}, 8 (1997), 145.   Google Scholar

[20]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, \emph{Funkcial. Ekvac.}, 40 (1997), 411.   Google Scholar

[21]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, \emph{RIMS Kokyuroku}, 1009 (1997), 22.   Google Scholar

[22]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, \emph{Journal. Theor. Biol.}, 42 (1973), 63.   Google Scholar

[23]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, \emph{Funkcial Ekvac}, 44 (2001), 441.   Google Scholar

[24]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 3727.  doi: 10.1016/j.nonrwa.2011.07.006.  Google Scholar

[25]

Z. A. Wang, Mathematics of traveling waves in chemotaxis,, \emph{Discrete Contin. Dyn. Syst.-Series B}, 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

[26]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, \emph{Journal of Differential Equations}, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[27]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, \emph{Mathematical Methods in the Applied Sciences}, 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

[28]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, \emph{Math. Nachr}, 283 (2010), 1664.  doi: 10.1002/mana.200810838.  Google Scholar

[29]

A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, \emph{Math. Jap}, 45 (1997), 241.   Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, \emph{Function Spaces, (1993), 9.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, \emph{Advances in Mathematical Sciences and Applications}, 9 (1999), 347.   Google Scholar

[3]

M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis,, \emph{Bioessays}, 28 (2006), 9.   Google Scholar

[4]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, \emph{Math. Bioscience}, 56 (1983), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[5]

D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation,, \emph{Current Opinion in Genetics Development}, 16 (2006), 367.   Google Scholar

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag-Berlin-New York, (1981).   Google Scholar

[7]

D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I,, \emph{Jahresber DMV}, 105 (2003), 103.   Google Scholar

[8]

D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II,, \emph{Jahresber DMV}, 106 (2003), 51.   Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis,, \emph{Journal of Mathematical Biology}, 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius,, \emph{Discrete Contin. Dyn. Syst-Series B}, 7 (2007), 125.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system,, \emph{J. Diff. Equation}, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model,, \emph{Journal of Mathematical Biology}, 35 (1996), 583.  doi: 10.1007/s002850050049.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability,, \emph{Journal of Theoratical Biology}, 26 (1970), 399.   Google Scholar

[14]

E. F. Keller and L. A. Segel, Model for Chemotaxis,, \emph{Journal of Theoratical Biology}, 30 (1971), 225.   Google Scholar

[15]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, \emph{Journal of Theoratical Biology}, 30 (1971), 235.   Google Scholar

[16]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, \emph{SIAM J. Appl. Math.}, 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[17]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, \emph{Journal of Mathematical Biology}, 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[19]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, \emph{Adv. Math. Soc. Appl.}, 8 (1997), 145.   Google Scholar

[20]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, \emph{Funkcial. Ekvac.}, 40 (1997), 411.   Google Scholar

[21]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, \emph{RIMS Kokyuroku}, 1009 (1997), 22.   Google Scholar

[22]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, \emph{Journal. Theor. Biol.}, 42 (1973), 63.   Google Scholar

[23]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, \emph{Funkcial Ekvac}, 44 (2001), 441.   Google Scholar

[24]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 3727.  doi: 10.1016/j.nonrwa.2011.07.006.  Google Scholar

[25]

Z. A. Wang, Mathematics of traveling waves in chemotaxis,, \emph{Discrete Contin. Dyn. Syst.-Series B}, 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

[26]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, \emph{Journal of Differential Equations}, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[27]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, \emph{Mathematical Methods in the Applied Sciences}, 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

[28]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, \emph{Math. Nachr}, 283 (2010), 1664.  doi: 10.1002/mana.200810838.  Google Scholar

[29]

A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, \emph{Math. Jap}, 45 (1997), 241.   Google Scholar

[1]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[2]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
[3]

Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821
[4]

Youshan Tao. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2705-2722. doi: 10.3934/dcdsb.2013.18.2705
[5]

Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
[6]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[7]

Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
[8]

Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[9]

Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239
[10]

Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262
[11]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[12]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[13]

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
[14]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[15]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[16]

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
[17]

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
[18]

Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[19]

Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777
[20]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences