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

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 $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.
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
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show all references

References:
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Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.  Google Scholar

[2]

Lect. 2nd 1985 Sess. C. I. M. E., Montecatini Terme/Italy 1985, Lect. Notes Math., 1224 (1986), 1-46. doi: 10.1007/BFb0072687.  Google Scholar

[3]

Dev. Biol., 296 (2006), 137-149. doi: 10.1016/j.ydbio.2006.04.451.  Google Scholar

[4]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437-446. doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

[5]

Banach Center Publ., 81 (2008), Polish Acad. Sci., Warsaw, 105-117. doi: 10.4064/bc81-0-7.  Google Scholar

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Holt, Rinehart & Winston, New York, 1969.  Google Scholar

[7]

Euro. J. Neuroscicen, 19 (2004), 831-844. Google Scholar

[8]

Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[10]

Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.  Google Scholar

[11]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

Jahresber. Deutsch. Math.- Verien, 105 (2003), 103-165.  Google Scholar

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European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.  Google Scholar

[14]

J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[15]

Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.  Google Scholar

[16]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[17]

J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[18]

Arch. Rational Mech. Anal., 74 (1980), 335-353. doi: 10.1007/BF00249679.  Google Scholar

[19]

Bull. Math. Biol., 65 (2003), 673-730. Google Scholar

[20]

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[21]

Can. Appl. Math. Quart., 10 (2002), 501-543.  Google Scholar

[22]

Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[23]

Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.  Google Scholar

[24]

J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[25]

J. Differential Equations, 252 (2012), 2520-2534. doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[26]

Ann. Inst. H. Poincaré, Analyse Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

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[28]

J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[29]

J. Math. Pures Appl., 99 (2013). doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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