Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species

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January 2016, 15(1): 243-260. doi: 10.3934/cpaa.2016.15.243

Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species

1.	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.	Department of Applied Mathematics, Dalian University of Technology, Dalian 116024

Received May 2015 Revised September 2015 Published December 2015

Abstract
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This paper deals with two-species quasilinear parabolic-parabolic Keller-Segel system $ u_{it}=\nabla\cdot(\phi_i(u_i)\nabla u_i)-\nabla\cdot(\psi_i(u_i)\nabla v)$, $i=1,2$, $v_t=\Delta v-v+u_1+u_2$ in $\Omega\times (0,T)$, subject to the homogeneous Neumann boundary conditions, with bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$. We prove that if $\frac{\psi_i(u_i)}{\phi_i(u_i)}\leq C_iu_i^{\alpha_i}$ for $u_i>1$ with $0<\alpha_i<\frac{2}{n}$ and $C_i>0$, $i=1,2$, then the solutions are globally bounded, while if $\frac{\psi_1(u_1)}{\phi_1(u_1)}\geq C_1u_1^{\alpha_1}$ for $u_1>1$ with $\Omega=B_R$, $\alpha_1>\frac{2}{n}$, then for any radial $u_{20}\in C^0(\overline{\Omega})$ and $m_1>0$, there exists positive radial initial data $u_{10}$ with $\int_\Omega u_{10}=m_1$ such that the solution blows up in a finite time $T_{\max}$ in the sense $\lim_{{t\rightarrow T_{\max}}} \|u_1(\cdot,t)+u_2(\cdot,t)\|_{L^{\infty}(\Omega)}=\infty$. In particular, if $\alpha_1>\frac{2}{n}$ with $0<\alpha_2<\frac{2}{n}$, the finite time blow-up for the species $u_1$ is obtained under suitable initial data, a new phenomenon unknown yet even for the semilinear Keller-Segel system of two species.

Keywords: chemotaxis, boundedness, blow-up., Two-species Keller-Segel system.

Mathematics Subject Classification: 92C17, 35K55, 35B4.

Citation: Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

References:

[1]	X. R. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source,, \emph{J. Math. Anal. Appl.}, 412 (2014), 181. doi: 10.1016/j.jmaa.2013.10.061.
[2]	T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, \emph{J. Differential Equations}, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045.
[3]	T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, \emph{Acta Appl. Math.}, 129 (2014), 135. doi: 10.1007/s10440-013-9832-5.
[4]	T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models,, \emph{J. Differential Equations}, 258 (2015), 2080. doi: 10.1016/j.jde.2014.12.004.
[5]	T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, \emph{Nonlinearity}, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.
[6]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, \emph{Ann. Scuola Normale Superiore}, 24 (1997), 633.
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, \emph{Jber. DMV}, 105 (): 103.
[8]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species,, \emph{J. Nonlinear Sci.}, 21 (2011), 231. doi: 10.1007/s00332-010-9082-x.
[9]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, \emph{J. Differential Equations}, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.
[10]	S. Ishida, K. Seki and T, Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, \emph{J. Differential Equations}, 256 (2014), 2993. doi: 10.1016/j.jde.2014.01.028.
[11]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, \emph{Trans. Amer. Math. Soc.}, 329 (1992), 819. doi: 10.2307/2153966.
[12]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, AMS, (1968).
[13]	Y. Li and Y. X. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species,, \emph{Nonlinear Anal.}, 109 (2014), 72. doi: 10.1016/j.na.2014.05.021.
[14]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, \emph{Can. Appl. Math. Q.}, 10 (2002), 501.
[15]	Y. S. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, \emph{J. Differential Equations}, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019.
[16]	M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, \emph{Math. Meth. Appl. Sci.}, 33 (2010), 12. doi: 10.1002/mma.1146.
[17]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, \emph{J. Differential Equations}, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008.
[18]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, \emph{J. Math. Pures Appl.}, 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020.
[19]	Q. S. Zhang and Y. X. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system,, \emph{J. Math. Anal. Appl.}, 418 (2014), 47. doi: 10.1016/j.jmaa.2014.03.084.

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References:

[1]	X. R. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source,, \emph{J. Math. Anal. Appl.}, 412 (2014), 181. doi: 10.1016/j.jmaa.2013.10.061.
[2]	T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, \emph{J. Differential Equations}, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045.
[3]	T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, \emph{Acta Appl. Math.}, 129 (2014), 135. doi: 10.1007/s10440-013-9832-5.
[4]	T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models,, \emph{J. Differential Equations}, 258 (2015), 2080. doi: 10.1016/j.jde.2014.12.004.
[5]	T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, \emph{Nonlinearity}, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.
[6]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, \emph{Ann. Scuola Normale Superiore}, 24 (1997), 633.
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, \emph{Jber. DMV}, 105 (): 103.
[8]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species,, \emph{J. Nonlinear Sci.}, 21 (2011), 231. doi: 10.1007/s00332-010-9082-x.
[9]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, \emph{J. Differential Equations}, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.
[10]	S. Ishida, K. Seki and T, Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, \emph{J. Differential Equations}, 256 (2014), 2993. doi: 10.1016/j.jde.2014.01.028.
[11]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, \emph{Trans. Amer. Math. Soc.}, 329 (1992), 819. doi: 10.2307/2153966.
[12]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, AMS, (1968).
[13]	Y. Li and Y. X. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species,, \emph{Nonlinear Anal.}, 109 (2014), 72. doi: 10.1016/j.na.2014.05.021.
[14]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, \emph{Can. Appl. Math. Q.}, 10 (2002), 501.
[15]	Y. S. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, \emph{J. Differential Equations}, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019.
[16]	M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, \emph{Math. Meth. Appl. Sci.}, 33 (2010), 12. doi: 10.1002/mma.1146.
[17]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, \emph{J. Differential Equations}, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008.
[18]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, \emph{J. Math. Pures Appl.}, 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020.
[19]	Q. S. Zhang and Y. X. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system,, \emph{J. Math. Anal. Appl.}, 418 (2014), 47. doi: 10.1016/j.jmaa.2014.03.084.

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