Large time behavior in a chemotaxis-Stokes system modeling coral fertilization with arbitrarily slow porous medium diffusion

This paper concerns with a kind of chemotaxis-Stokes systems generalizing the prototype {n_t+u⋅∇n=∇⋅(n^m−1∇n)−∇⋅(n∇c)−nv,c_t+u⋅∇c=Δc−c+v,v_t+u⋅∇v=Δv−nv,u_t=Δu+∇P+(n+v)∇Φ,∇⋅u=0 which characterizes the process of coral fertilization in ocean. By virtue of a novel approach on the basis of some conditional estimates for signal gradient and fluid velocity, it is proved that when m>1 an associated initial-boundary problem possesses a globally bounded weak solution in spatially three-dimensional setting, which extends the corresponding results obtained in [15]. Moreover, the obtained solutions stabilize to a certain constant equilibrium (n_∞,v_∞,v_∞,0) with [Formula presented] and [Formula presented] as t→∞.