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July/August 2018 Global stability in a two-competing-species chemotaxis system with two chemicals
Chunlai M, Yongsheng Mi, Pan Zheng
Differential Integral Equations 31(7/8): 547-558 (July/August 2018).

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.

Citation

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Chunlai M. Yongsheng Mi. Pan Zheng. "Global stability in a two-competing-species chemotaxis system with two chemicals." Differential Integral Equations 31 (7/8) 547 - 558, July/August 2018.

Information

Published: July/August 2018
First available in Project Euclid: 11 May 2018

zbMATH: 06890404
MathSciNet: MR3801824

Subjects:
Primary: 35B35, 35B40, 35K55, 92C17

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.31 • No. 7/8 • July/August 2018
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