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December  2013, 18(10): 2705-2722. doi: 10.3934/dcdsb.2013.18.2705

Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity

Youshan Tao 1,

1. 

Department of Applied Mathematics, Dong Hua University, Shanghai 200051

Received  November 2012 Revised  March 2013 Published  October 2013

This paper deals with the repulsion chemotaxis system $$ \left\{ \begin{array}{ll} u_t=\Delta u +\nabla \cdot (f(u)\nabla v), & x\in\Omega, \ t>0, \\ v_t=\Delta v +u-v, & x\in\Omega, \ t>0, \end{array} \right. $$ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ with $n\ge 3$, where $f(u)$ is the density-dependent chemotactic sensitivity function satisfying $$ f \in C^2([0, \infty)),      f(0)=0, 0 < f(u) \le K(u+1)^{\alpha}          for     all     u > 0 $$ with some $K>0$ and $\alpha>0$.
    It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.
Keywords: repulsion, Chemotaxis, nonlinear sensitivity, stabilization., boundedness.
Mathematics Subject Classification: Primary: 35A01, 35K51, 35K57, 35M33; Secondary: 35Q92, 92C1.
Citation: 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
References:
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N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827.  doi: 10.1080/03605307908820113.  Google Scholar

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M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit,, Euro. J. Neuroscicen, 19 (2004), 831.   Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977).   Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar

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

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T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

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D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103.   Google Scholar

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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

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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

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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: 10.2307/2153966.  Google Scholar

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

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R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

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P. L. Lions, Résolution de problèmes elliptiques quasilinéaires,, Arch. Rational Mech. Anal., 74 (1980), 335.  doi: 10.1007/BF00249679.  Google Scholar

[19]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and alzheimer's disease senile plague: is there a connection?, Bull. Math. Biol., 65 (2003), 673.   Google Scholar

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

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K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar

[22]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion-existence and branching instabilities,, Nonlinearity, 24 (2011), 1253.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[23]

Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar

[24]

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

[25]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[26]

Y. Tao and M. Winkler, Locally bounded global soutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion,, Ann. Inst. H. Poincaré, 30 (2013), 157.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[27]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911.  doi: 10.1002/mma.319.  Google Scholar

[28]

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

[29]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 99 (2013).  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

D. G. Aronson, The porous medium equation. Nonlinear diffusion problems,, Lect. 2nd 1985 Sess. C. I. M. E., 1224 (1986), 1.  doi: 10.1007/BFb0072687.  Google Scholar

[3]

M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. A. Glazier and C. J. Weijer, Cell movement during chick primitive streak formation,, Dev. Biol., 296 (2006), 137.  doi: 10.1016/j.ydbio.2006.04.451.  Google Scholar

[4]

T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437.  doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

[5]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar

[6]

A. Friedman, Partial Differential Equations,, Holt, (1969).   Google Scholar

[7]

M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit,, Euro. J. Neuroscicen, 19 (2004), 831.   Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977).   Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar

[10]

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

[11]

T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103.   Google Scholar

[13]

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

[14]

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

[15]

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: 10.2307/2153966.  Google Scholar

[16]

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

[17]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[18]

P. L. Lions, Résolution de problèmes elliptiques quasilinéaires,, Arch. Rational Mech. Anal., 74 (1980), 335.  doi: 10.1007/BF00249679.  Google Scholar

[19]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and alzheimer's disease senile plague: is there a connection?, Bull. Math. Biol., 65 (2003), 673.   Google Scholar

[20]

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

[21]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar

[22]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion-existence and branching instabilities,, Nonlinearity, 24 (2011), 1253.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[23]

Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar

[24]

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

[25]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[26]

Y. Tao and M. Winkler, Locally bounded global soutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion,, Ann. Inst. H. Poincaré, 30 (2013), 157.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[27]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911.  doi: 10.1002/mma.319.  Google Scholar

[28]

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

[29]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 99 (2013).  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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