American Institute of Mathematical Sciences

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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

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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

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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

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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

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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

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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

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