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Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

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  • This paper deals with the two-species chemotaxis-competition system

    $\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} \ Ω × (0, ∞), \\ v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} \ Ω × (0, ∞), \\ w_t = d_3Δ w + α u + β v - γ w&{\rm in} \ Ω × (0, ∞), \end{cases} \end{equation*}$

    where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n≥ 2$; $χ_i$ are functions satisfying some conditions. About this problem, Bai-Winkler [1] first obtained asymptotic stability in (1) under some conditions in the case that $a_1, a_2∈ (0, 1)$. Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that $a_1, a_2∈ (0, 1)$.

    Mathematics Subject Classification: Primary: 35K51; Secondary: 92C17, 35B40.

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    [2] T. BlackJ. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.
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    [6] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.
    [7] M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.
    [8] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.  doi: 10.1016/j.jde.2016.05.008.
    [9] M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.
    [10] M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.
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