Global boundedness and decay property of a three-dimensional Keller--Segel--Stokes system modeling coral fertilization

This paper is concerned with the four-component Keller–Segel–Stokes system modelling the fertilization process of corals: \begin{equation*} \left{ \begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho$\mathcal{S}$(x,\rho,c)\nabla c)-\rho m, & \quad (x,t)\in \Omega\times (0,T), \ m_t+u\cdot\nabla m=\Delta m-\rho m, & \quad (x,t)\in \Omega\times (0,T), \ c_t+u\cdot\nabla c=\Delta c-c+m, & \quad (x,t)\in \Omega\times (0,T), \ u_t=\Delta u-\nabla P+(\rho+m)\nabla\phi,\quad \nabla\cdot u=0, & \quad (x,t)\in \Omega\times (0,T) \end{array}\right. \end{equation*} subject to the boundary conditions $\nabla c\cdot \nu=\nabla m\cdot \nu=(\nabla\rho-\rho $\mathcal{S}$(x,\rho,c)\nabla c)\cdot \nu=0$and$u=0$,andsuitablyregularinitialdata$(\rho_0(x),m_0(x), c_0(x),u_0(x))$,where$T\in (0,\infty]$,$\Omega\subset\mathbb R^3$ is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. This system describes the spatio-temporal dynamics of the population densities of sperm $\rho$ and egg $m$ under a chemotactic process facilitated by a chemical signal released by the egg with concentration $c$ in a fluid-flow environment $u$ modeled by the incompressible Stokes equation. In this model, the chemotactic sensitivity tensor $\mathcal{S}\in C^2(\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty }{)}^2{)}^{3\times 3}$ satisfies $\mathrm{\mid }\mathcal{S}(x,\rho ,c)\mathrm{\mid }\le C_S(1+\rho {)}^{-\alpha }$ with some $C_S>0$ and $\alpha \ge 0$. We will show that for $\alpha \ge \frac{1}{3}$, the solutions to the system are globally bounded and decay to a spatially homogeneous equilibrium exponentially as time goes to infinity. In addition, we will also show that, for any $\alpha \ge 0$, a similar result is valid when the initial data satisfy a certain smallness condition.