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Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source. (English) Zbl 1363.35195

Summary: We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of X. Cao [J. Math. Anal. Appl. 412, No. 1, 181–188 (2014; Zbl 1364.35123)].

MSC:

35K59 Quasilinear parabolic equations
92C17 Cell movement (chemotaxis, etc.)

Citations:

Zbl 1364.35123
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References:

[1] X. Cao: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source. J. Math. Anal. Appl. 412 (2014), 181–188. · Zbl 1364.35123 · doi:10.1016/j.jmaa.2013.10.061
[2] T. Ciešlak, C. Stinner: Finite-time blowup in a supercritical quasilinear parabolicparabolic Keller-Segel system in dimension 2. Acta Appl. Math. 129 (2014), 135–146. · Zbl 1295.35123 · doi:10.1007/s10440-013-9832-5
[3] T. Ciešlak, C. Stinner: Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions. J. Differ. Equations 252 (2012), 5832–5851. · Zbl 1252.35087 · doi:10.1016/j.jde.2012.01.045
[4] M. A. Herrero, J. J. L. Velázquez: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4. 24 (1997), 633–683. · Zbl 0904.35037
[5] D. Horstmann, G. Wang: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12 (2001), 159–177. · Zbl 1017.92006 · doi:10.1017/S0956792501004363
[6] D. Horstmann, M. Winkler: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equations 215 (2005), 52–107. · Zbl 1085.35065 · doi:10.1016/j.jde.2004.10.022
[7] E. F. Keller, L. A. Segel: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399–415. · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[8] T. Nagai, T. Senba, K. Yoshida: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj. Ser. Int. 40 (1997), 411–433. · Zbl 0901.35104
[9] K. Osaki, A. Yagi: Finite dimensional attractor for one-dimensional Keller-Segel equations. Funkc. Ekvacioj. Ser. Int. 44 (2001), 441–469. · Zbl 1145.37337
[10] K. J. Painter, T. Hillen: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10 (2002), 501–543. · Zbl 1057.92013
[11] Y. Tao, M. Winkler: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differ. Equations 252 (2012), 692–715. · Zbl 1382.35127 · doi:10.1016/j.jde.2011.08.019
[12] M. Winkler: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J. Math. Pures Appl. 100 (2013), 748–767. · Zbl 1326.35053 · doi:10.1016/j.matpur.2013.01.020
[13] M. Winkler: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equations 248 (2010), 2889–2905. · Zbl 1190.92004 · doi:10.1016/j.jde.2010.02.008
[14] M. Winkler: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equations 35 (2010), 1516–1537. · Zbl 1290.35139 · doi:10.1080/03605300903473426
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