Research article

Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system

  • Received: 21 July 2016 Accepted: 03 August 2016 Published: 10 August 2016
  • This paper deals with the two-species chemotaxis system $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{v_t} = \Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - v)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{w_t} = d\Delta w + h(u,v,w)}&{{\text{in}}\;\Omega \times (0,\infty ),} \end{array}} \right.$ where $\Omega$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, $N\in \mathbb{N}$; $h$, $\chi_i$ are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.

    Citation: Masaaki Mizukami. Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system[J]. AIMS Mathematics, 2016, 1(3): 156-164. doi: 10.3934/Math.2016.3.156

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  • This paper deals with the two-species chemotaxis system $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{v_t} = \Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - v)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{w_t} = d\Delta w + h(u,v,w)}&{{\text{in}}\;\Omega \times (0,\infty ),} \end{array}} \right.$ where $\Omega$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, $N\in \mathbb{N}$; $h$, $\chi_i$ are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.
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    [11] M. Mizukami, T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diflusion. J. Diflerential Equations 261 (2016), 2650-2669.
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    © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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