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
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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

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