Boundedness and stabilization in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Jin, Hai-Yang¹  Liu, Zhengrong¹  Shi, Shijie²  Xu, Jiao¹ 

[1] South China Univ Technol, Sch Math, Guangzhou 510640, Guangdong, Peoples R China

[2] Shenzhen Technol Univ, Fac Arts & Sci, Shenzhen 518118, Peoples R China

关键词: Chemotaxis;    Boundedness;    Stabilization;    Signal-dependent diffusion;    Signal-dependent sensitivity;   

DOI  :  10.1016/j.jde.2019.01.019

来源: Elsevier

【 摘 要 】

We consider the following two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity {u(1t) = del . (d(1)(v)del u(1)) - del. (chi(1)(v)u(1)del v) + mu(1)u(1) (1 - u(1) - a(1)u(2)), x is an element of Omega, t > 0, u(2t) = del . (d(2)(v)del u(2)) - del. (chi(2)(v)u(2)del v) + mu(2)u(2) (1 - u(2) - a(2)u(1)), x is an element of Omega, t > 0, v(t) = Delta v + b(1)u(1) + b(2)u(2) - v, x is an element of Omega, t > 0, u(1) (x, 0) = u(10) (x), u(2)(x, 0) = u(20)(x), v(x, 0) = v(0)(x), x is an element of Omega, in a bounded smooth domain Omega subset of R-2 with homogeneous Neumann boundary conditions, where mu(i), a(i), b(i) are positive constants for i = 1, 2, and the functions d(i) (v), chi(i) (v) satisfy the following assumptions: (d(i) (v), chi(i) (v)) is an element of [C-2[0, infinity)](2) with d(i) (v), chi(i) (v) > 0 for all v >= 0, d(i)(1) (v) < 0 and v ->infinity lim d(i) (v) = 0; v ->infinity lim chi(i) (v)/d(i) (v) and v ->infinity lim d(i)'(v)/d(i) (v) exist. Since v ->infinity lim d(i) (v) = 0 for i = 1, 2, the diffusion may degenerate, which makes the analysis of system (*) much more difficult. To overcome this problem, we shall use the functions d(i) (v) as weight functions and then employ the weighted energy estimates to establish the boundedness of solutions. Furthermore, by constructing some appropriate Lyapunov functionals, we show that If a(1), a(2) is an element of (0, 1) and mu(1), mu(2) are large enough, then the solution (u(1), u(2), v) exponentially converges to (1-a(1)/1-a(1)a(2), 1-a(2)/1-a(1)a(2) , b(1)+b(2)-a(1)b(1)-a(2)b(2)/1-a(1)a(2)) as t -> infinity. If a(1) >= 1, a(2) is an element of (0, 1) and mu(2) is large enough, the solution (u(1), u(2), v) converges to (0, 1, b(2)) as t -> infinity with algebraic decay when a(1) = 1, and with exponential decay when a(1) > 1. (C) 2019 Elsevier Inc. All rights reserved.

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