Asymptotic Behavior in a Quasilinear Two-Species Chemotaxis System with Nonlinear Sensitivity and Nonlinear Signal Secretion

Abstract

In this paper, we study the asymptotic behavior of solutions to a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal production

$\left\{\begin{array}{ll}u_{t}=\nabla \cdot\left(D_{1}(u) \nabla u\right)-\nabla \cdot\left(S_{1}(u) \nabla v\right)+\mu_{1} u\left(1-u-a_{1} w\right), & x \in \Omega, t>0, \\\tau v_{t}=\Delta v-v+w^{\gamma_{1}}, & x \in \Omega, t>0, \\w_{t}=\nabla \cdot\left(D_{2}(w) \nabla w\right)-\nabla \cdot\left(S_{2}(w) \nabla z\right)+\mu_{2} w\left(1-w-a_{2} u\right), & x \in \Omega, t>0, \\\tau z_{t}=\Delta z-z+u^{\gamma_{2}}, & x \in \Omega, t>0\end{array}\right.$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset \mathbb{R}^{n}(n \geq 2)$ where the parameter $\mu_{1}, \mu_{2}, \gamma_{1}, \gamma_{2}$ are positive constants, $\tau \in\{0,1\}$. The diffusion coefficients $D_{i}, S_{i} \in C^{2}([0, \infty))$, satisfy $D_{i}(s) \geq a_{0}(s+1)^{-m_{i}}$, $0 \leq S_{i}(s) \leq b_{0} s(s+1)^{\beta_{i}-1}$, $s \geq 0$, $m_{i}, \beta_{i} \in \mathbb{R}, a_{0}, b_{0}>0, i=1,2$. Under the assumption of properly initial data regularity, we can find appropriate μiμi such that the globally bounded solution of this system satisfies the following relationship.

(I) If $a_{1}, a_{2} \in(0,1)$ and μ₁ and μ₂ are sufficiently large, then any globally bounded solution exponentially converges to $\left(\frac{1-a_{1}}{1-a_{1} a_{2}},\left(\frac{1-a_{2}}{1-a_{1} a_{2}}\right)^{\gamma_{1}}, \frac{1-a_{2}}{1-a_{1} a_{2}},\left(\frac{1-a_{1}}{1-a_{1} a_{2}}\right)^{\gamma_{2}}\right)$ as t→∞;

(II) If $a_{1}>1>a_{2}>0$ and μ₂ is sufficiently large, then any globally bounded solution exponentially converges to (0,1,1,0) as t→∞;

(III) If $a_{1}=1>a_{2}>0$ and μ₂ is sufficiently large, then any globally bounded solution algebraically converges to (0,1,1,0) as t→∞.

Keywords

Two-species chemotaxis / asymptotic behavior / exponential convergence / algebraical convergence