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Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. (English) Zbl 1442.35478

Summary: This paper deals with the two-species chemotaxis-competition system \[\begin{cases} u_t = d_1 \Delta u - \nabla \cdot (u \chi_1(w)\nabla w) + \mu_1 u(1-u-a_1 v)\quad &\text{in} \Omega \times (0, \infty), \\ v_t = d_2 \Delta v - \nabla \cdot (v \chi_2(w)\nabla w) + \mu_2 v(1-a_2u-v)\quad &\text{in} \Omega \times (0, \infty ), \\ w_t = d_3 \Delta w + \alpha u + \beta v - \gamma w \quad &\text{in} \Omega \times (0, \infty), \end{cases}\] where \(\Omega\) is a bounded domain in \( \mathbb{R}^n\) with smooth boundary \( \partial \Omega\), \( n \geq 2\); \(\chi_i\) are functions satisfying some conditions. About this problem, X. Bai and M. Winkler [Indiana Univ. Math. J. 65, No. 2, 553–583 (2016; Zbl 1345.35117)] first obtained asymptotic stability in (1) under some conditions in the case that \(a_1\), \(a_2 \in (0, 1) \). Recently, the conditions assumed in [loc. cit.] were improved [M. Mizukami, Discrete Contin. Dyn. Syst., Ser. B 22, No. 6, 2301–2319 (2017; Zbl 1366.35068)]; however, there is a gap between the conditions assumed in [Bai and Winkler, loc. cit.] and [Mizukami, loc. cit.]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that \(a_1\), \(a_2 \in (0, 1) \).

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C17 Cell movement (chemotaxis, etc.)
35K51 Initial-boundary value problems for second-order parabolic systems
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

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