## Differential and Integral Equations

- Differential Integral Equations
- Volume 31, Number 7/8 (2018), 547-558.

### Global stability in a two-competing-species chemotaxis system with two chemicals

Chunlai M, Yongsheng Mi, and Pan Zheng

#### Abstract

This paper deals with a two-competing-species chemotaxis system with two different chemicals \begin{equation*} \begin{cases} u_{t}=d_{1}\Delta u-\chi_{1}\nabla \cdot(u\nabla v)+\mu_{1} u(1-u-a_{1}w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{2}\Delta v-\alpha_{1}v+\beta_{1}w, & (x,t)\in \Omega\times (0,\infty),\\ w_{t}=d_{3}\Delta w-\chi_{2}\nabla \cdot(w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{4}\Delta z-\alpha_{2}z+\beta_{2}u, & (x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$ with nonnegative initial data $(u_{0},w_{0})\in (C^{0}(\overline{\Omega}))^{2}$ satisfying $u_{0}\not\equiv0$ and $w_{0}\not\equiv 0$, where $\chi_{1},\chi_{2}\geq0$, $a_{1}, a_{2}\in[0,1)$, and the parameters $d_{i}$ ($i=1,2,3,4$) and $\alpha_{j},\beta_{j}, \mu_{j}$ ($j=1,2$) are positive. Based on the approach of eventual comparison, it is shown that under suitable conditions, the system possesses a unique global-in-time classical solution, which converges to the constant steady states.

#### Article information

**Source**

Differential Integral Equations, Volume 31, Number 7/8 (2018), 547-558.

**Dates**

First available in Project Euclid: 11 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1526004030

**Mathematical Reviews number (MathSciNet)**

MR3801824

**Zentralblatt MATH identifier**

06890404

**Subjects**

Primary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations 92C17: Cell movement (chemotaxis, etc.)

#### Citation

Zheng, Pan; M, Chunlai; Mi, Yongsheng. Global stability in a two-competing-species chemotaxis system with two chemicals. Differential Integral Equations 31 (2018), no. 7/8, 547--558. https://projecteuclid.org/euclid.die/1526004030