# American Institute of Mathematical Sciences

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

 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.
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. [2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, \emph{Advances in Mathematical Sciences and Applications}, 9 (1999), 347. [3] M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis,, \emph{Bioessays}, 28 (2006), 9. [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. [5] D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation,, \emph{Current Opinion in Genetics Development}, 16 (2006), 367. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag-Berlin-New York, (1981). [7] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I,, \emph{Jahresber DMV}, 105 (2003), 103. [8] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II,, \emph{Jahresber DMV}, 106 (2003), 51. [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. [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. [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. [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. [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. [14] E. F. Keller and L. A. Segel, Model for Chemotaxis,, \emph{Journal of Theoratical Biology}, 30 (1971), 225. [15] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, \emph{Journal of Theoratical Biology}, 30 (1971), 235. [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. [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. [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968). [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. [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. [21] T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, \emph{RIMS Kokyuroku}, 1009 (1997), 22. [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, \emph{Journal. Theor. Biol.}, 42 (1973), 63. [23] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, \emph{Funkcial Ekvac}, 44 (2001), 441. [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. [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. [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. [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. [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. [29] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, \emph{Math. Jap}, 45 (1997), 241.

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##### 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. [2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, \emph{Advances in Mathematical Sciences and Applications}, 9 (1999), 347. [3] M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis,, \emph{Bioessays}, 28 (2006), 9. [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. [5] D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation,, \emph{Current Opinion in Genetics Development}, 16 (2006), 367. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag-Berlin-New York, (1981). [7] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I,, \emph{Jahresber DMV}, 105 (2003), 103. [8] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II,, \emph{Jahresber DMV}, 106 (2003), 51. [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. [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. [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. [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. [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. [14] E. F. Keller and L. A. Segel, Model for Chemotaxis,, \emph{Journal of Theoratical Biology}, 30 (1971), 225. [15] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, \emph{Journal of Theoratical Biology}, 30 (1971), 235. [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. [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. [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968). [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. [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. [21] T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, \emph{RIMS Kokyuroku}, 1009 (1997), 22. [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, \emph{Journal. Theor. Biol.}, 42 (1973), 63. [23] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, \emph{Funkcial Ekvac}, 44 (2001), 441. [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. [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. [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. [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. [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. [29] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, \emph{Math. Jap}, 45 (1997), 241.
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