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: {rho(t) + u . del rho = Delta rho - del . (rho S(x, rho, c)del c) - rho m, (x, t) is an element of Omega x (0, T), m(t) + u . del m = del m - rho m, (x, t) is an element of Omega x (0, T), c(t) + u . del c = Delta c - c + m, (x, t) is an element of Omega x (0, T), u(t) = Delta u - del P + (rho + m) del phi, del . u = 0, (x, t) is an element of Omega x (0, T) subject to the boundary conditions del c . nu = del m . nu = (del rho - rho S(x, rho, c) del c) . nu = 0 and u = 0, and suitably regular initial data (rho(0)(x), m(0)(x), c(0)(x), u(0)(x)), where T is an element of (0, infinity], Omega subset of R-3 is a bounded domain with smooth boundary partial derivative 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 S is an element of C-2((Omega) over bar x [0, infinity)(2))(3x3) satisfies vertical bar S(x, rho, c)vertical bar <= C-S(1 + rho)(-alpha) with some C-S > 0 and alpha >= 0. We will show that for alpha >= 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 >= 0, a similar result is valid when the initial data satisfy a certain smallness condition.