Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions

Abstract

This paper studies the two-species chemotaxis system with two indirect signal production and logistic source $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} u_t=\Delta u-\chi _{1}\nabla \cdot (u\nabla v)+\mu _{1}u(1-u),\quad &{}x\in \Omega ,\quad t>0,\\ v_{t} =\Delta v-v+w,\quad &{}x\in \Omega ,\quad t>0,\\ w_{t} =\Delta w-\chi _{2}\nabla \cdot (w\nabla z)+\mu _{2}w(1-w),\quad &{}x\in \Omega ,\quad t>0,\\ z_{t}=\Delta z-z+u,\quad &{}x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$ in a bounded smooth domain \(\Omega \subset \mathbb {R}^3\) with zero-flux boundary conditions, where the parameters \(\chi _{i}\) and \(\mu _{i}\)\((i=1,2)\) are positive. It is shown that this system possesses a unique global-bounded classical solution under the conditions \(\mu _{1}\ge \max \Big \{7\chi _{1}^{2}+1,\frac{51}{2}\Big \}\) and \(\mu _{2}\ge \max \Big \{7\chi _{2}^{2}+1,\frac{51}{2}\Big \}\). Furthermore, whenever \(u_{0}\not \equiv 0\), the solution of the system exponentially stabilizes to the constant stationary solution \(\big (1,1,1,1\big )\) in the norm of \(L^{\infty }(\Omega )\) as \(t\rightarrow \infty \) under the additional conditions \(\mu _{1}>\frac{\chi _{2}^{2}}{8}\) and \(\mu _{2}>\frac{\chi _{1}^{2}}{8}\).