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

Miaoqing  Tian; Sining Zheng

doi:10.3934/cpaa.2016.15.243

Article Contents

2016, Volume 15, Issue 1: 243-260. Doi: 10.3934/cpaa.2016.15.243

This issue Previous Article On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior Next Article Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators

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

Miaoqing Tian^1, and
Sining Zheng^2,

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

Received: April 30, 2015

Revised: August 31, 2015

Published: December 2015

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 92C17, 35K55, 35B40.

Citation:

Related Papers

Cited by

References

[1]	X. R. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.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, J. Differential Equations, 252 (2012), 5832-5851.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, Acta Appl. Math., 129 (2014), 135-146.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, J. Differential Equations, 258 (2015), 2080-2113.doi: 10.1016/j.jde.2014.12.004.
[5]	T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.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, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jber. DMV, 105, 103-165.
[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, J. Nonlinear Sci., 21 (2011), 231-270.doi: 10.1007/s00332-010-9082-x.
[9]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.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, J. Differential Equations, 256 (2014), 2993-3010.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, Trans. Amer. Math. Soc., 329 (1992), 819-824.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, Providence, 1968.
[13]	Y. Li and Y. X. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.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, Can. Appl. Math. Q., 10 (2002), 501-543.
[15]	Y. S. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019.
[16]	M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse? Math. Meth. Appl. Sci., 33 (2010), 12-24.doi: 10.1002/mma.1146.
[17]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.doi: 10.1016/j.jde.2010.02.008.
[18]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.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, J. Math. Anal. Appl., 418 (2014), 47-63.doi: 10.1016/j.jmaa.2014.03.084.