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:subject to the boundary conditions and u = 0, and suitably regular initial data , where , is a bounded domain with smooth boundary . This system describes the spatio-temporal dynamics of the population densities of sperm 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 satisfies with some CS > 0 and . We will show that for , 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 , a similar result is valid when the initial data satisfy a certain smallness condition.