Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals

Youshan Tao; Michael Winkler

doi:10.3934/dcdsb.2015.20.3165

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November 2015, 20(9): 3165-3183. doi: 10.3934/dcdsb.2015.20.3165

Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals

Youshan Tao ^1, and Michael Winkler ^2,

1.	Department of Applied Mathematics, Dong Hua University, Shanghai 200051
2.	Institut für Mathematik, Universität Paderborn, 33098 Paderborn

Received August 2014 Revised April 2015 Published September 2015

Abstract
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We consider a model for two species interacting through chemotaxis in such a way that each species produces a signal which directs the respective motion of the other. Specifically, we shall be concerned with nonnegative solutions of the Neumann problem, posed in bounded domains $\Omega\subset \mathbb{R}^n$ with smooth boundary, for the system $$\begin{cases} u_t= \Delta u - \chi \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v-v+w, & x\in \Omega, \, t>0, \qquad (\star)\\ w_t= \Delta w - \xi \nabla \cdot (w\nabla z), & x\in \Omega, \, t>0, \\ 0=\Delta z-z+u, & x\in \Omega, \, t>0, \end{cases}$$
with parameters $\chi \in \{\pm 1\}$ and $\xi\in \{\pm 1\}$, thus allowing the interaction of either attraction-repulsion, or attraction-attraction, or repulsion-repulsion type.
It is shown that
$\bullet$ in the attraction-repulsion case $\chi=1$ and $\xi=-1$, if $n\le 3$ then for any nonnegative initial data $u_0\in C^0(\bar{\Omega})$ and $ w_0\in C^0 (\bar{\Omega})$, there exists a unique global classical solution which is bounded;
$\bullet$ in the doubly repulsive case when $\chi=\xi=-1$, the same holds true;
$\bullet$ in the attraction-attraction case $\chi=\xi=1$,
$-$ if either $n=2$ and $\int_\Omega u_0 + \int_\Omega w_0$ lies below some threshold, or $n\ge 3$ and $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$ are sufficiently small, then solutions exist globally and remain bounded, whereas
$-$ if either $n=2$ and $m$ is suitably large, or $n\ge 3$ and $m>0$ is arbitrary, then there exist smooth initial data $u_0$ and $w_0$ such that $\int_\Omega u_0 + \int_\Omega w_0=m$ and such that the corresponding solution blows up in finite time.
In particular, these results demonstrate that the circular chemotaxis mechanism underlying ($\star$) goes along with essentially the same destabilizing features as known for the classical Keller-Segel system in the doubly attractive case, but totally suppresses any blow-up phenomenon when only one, or both, taxis directions are repulsive.

Keywords: preventing blow-up., repulsion, attraction, Chemotaxis.

Mathematics Subject Classification: Primary: 35A01, 35B44, 35K57, 35Q92, 92C1.

Citation: Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165