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

Abstract

This paper is concerned with the four-component Keller--Segel--Stokes system modelling the fertilization process of corals: \begin{equation*} \left\{ \begin{array}{ll} ρ_t+u\cdot\nablaρ=Δρ-\nabla\cdot(ρ\mathcal{S}(x,ρ,c)\nabla c)-ρm, & \quad (x,t)\in Ω\times (0,T), \\ m_t+u\cdot\nabla m=Δm-ρm, & \quad (x,t)\in Ω\times (0,T), \\ c_t+u\cdot\nabla c=Δc-c+m, & \quad (x,t)\in Ω\times (0,T), \\ u_t=Δu-\nabla P+(ρ+m)\nablaϕ,\quad \nabla\cdot u=0, & \quad (x,t)\in Ω\times (0,T) \end{array}\right. \end{equation*} subject to the boundary conditions $\nabla c\cdot ν=\nabla m\cdot ν=(\nablaρ-ρ\mathcal{S}(x,ρ,c)\nabla c)\cdot ν=0$ and $u=0$, and suitably regular initial data $(ρ_0(x),m_0(x), c_0(x),u_0(x))$, where $T\in (0,\infty]$, $Ω\subset\mathbb R^3$ is a bounded domain with smooth boundary $\partialΩ$. This system describes the spatio-temporal dynamics of the population densities of sperm $ρ$ 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 $\mathcal{S}\in C^2(\overlineΩ\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,ρ,c)|\leq C_S(1+ρ)^{-α}$ with some $C_S>0$ and $α\geq 0$. We will show that for $α\geq \frac 13$, 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 $α\geq 0$, a similar result is valid when the initial data satisfy a certain smallness condition.