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June  2016, 21(4): 1317-1327. doi: 10.3934/dcdsb.2016.21.1317

Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity

Hao Yu 1, , Wei Wang 1, and  Sining Zheng 1,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China, China

Received  July 2015 Revised  November 2015 Published  March 2016

This paper considers the parabolic-parabolic Keller-Segel system with nonlinear sensitivity $u_t=\Delta u-\nabla (u^{\alpha}\nabla v)$, $v_t=\Delta v-v+u$, subject to homogeneous Neumann boundary conditions with smooth and bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq1$. It is proved that if $\alpha\geq\max\{1,\frac{2}{n}\}$, then the solutions are globally bounded, and both the components $u$ and $v$ decay to the same constant steady state $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0(x) dx$ exponentially in the $L^\infty$-norm provided both $\|u_0\|_{L^{q^{\ast}}(\Omega)}$ and $\|\nabla v_0\|_{L^{p^{\ast}}(\Omega)}$ small enough with $q^{\ast}=\frac{n\alpha K}{n+K}$, $p^{\ast}=\frac{n\alpha K}{n+K-n\alpha}$, $K\in[n,2n\alpha-n]\cap ((\alpha-1)n,\infty)$.
Keywords: Chemotaxis, nonlinear sensitivity., Keller-Segel system, boundedness.
Mathematics Subject Classification: Primary: 35K55, 35B35, 35B40; Secondary: 92C1.
Citation: Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
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T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar

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

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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

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M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

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show all references

References:
[1]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.   Google Scholar

[2]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces,, Discrete Contin. Dyn. Syst., 35 (2015), 1891.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[3]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[4]

T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135.  doi: 10.1007/s10440-013-9832-5.  Google Scholar

[5]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models,, J. Differential Equations, 258 (2015), 2080.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

[6]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159.  doi: 10.1017/S0956792501004363.  Google Scholar

[9]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[12]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[13]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

[19]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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