Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 197–220

# Blow up and Bounded Solutions in a Two-Species Chemotaxis System in Two Dimensional Domains

Article

## Abstract

In this paper, we consider the initial-boundary value problem of the two-species chemotaxis Keller-Segel model
\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla w), &x\in \varOmega , \ t>0, \\ v_{t}=\Delta v-\chi_{2}\nabla \cdot (v\nabla w), &x\in \varOmega , \ t>0, \\ 0=\Delta w-\gamma w+\alpha_{1}u+\alpha_{2}v, &x\in \varOmega , \ t>0, \end{cases}\displaystyle \end{aligned}
where the parameters $$\chi_{1}$$, $$\chi_{2}$$, $$\alpha_{1}$$, $$\alpha_{2}$$, $$\gamma$$ are positive constants, $$\varOmega \subset \mathbb{R}^{2}$$ is a bounded domain with smooth boundary. We obtain the results for finite time blow-up and global bounded as follows: (1) For any fixed $$x_{0}\in \varOmega$$, if $$\chi_{1}\alpha_{2}= \chi_{2}\alpha_{1}$$, $$\int_{\varOmega }(u_{0}+v_{0})|x-x_{0}|^{2}dx$$ is sufficiently small, and $$\int_{\varOmega }(u_{0}+v_{0})dx>\frac{8\pi ( \chi_{1}\alpha_{1}+\chi_{2}\alpha_{2})}{\chi_{1}\alpha_{1}\chi_{2} \alpha_{2}}$$, then the nonradial solution of the two-species Keller-Segel model blows up in finite time. Moreover, if $$\varOmega$$ is a convex domain, we find a lower bound for the blow-up time; (2) If $$\|u_{0}\|_{L^{1}(\varOmega )}$$ and $$\|v_{0}\|_{L^{1}( \varOmega )}$$ lie below some thresholds, respectively, then the solution exists globally and remains bounded.

### Keywords

Chemotaxis Blow-up Boundedness

### Mathematics Subject Classification

92C17 35B44 39A22

## Notes

### Acknowledgements

The authors are very grateful to the anonymous reviewers for their careful reading and valuable comments which greatly improved this work. This work is supported by NSFC (Grant No. 11371384 and No. 11571062) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007).

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## Authors and Affiliations

• Jie Zhao
• 1
• Chunlai Mu
• 1
• Liangchen Wang
• 1
• Deqin Zhou
• 1
1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingP.R. China