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

Abstract

This paper concerns with a kind of chemotaxis-Stokes systems generalizing the prototype{nt+u⋅∇n=∇⋅(nm−1∇n)−∇⋅(n∇c)−nv,ct+u⋅∇c=Δc−c+v,vt+u⋅∇v=Δv−nv,ut=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 n∞:=1|Ω|{∫Ωn0−∫Ωv0}+ and v∞:=1|Ω|{∫Ωv0−∫Ωn0}+ as t→∞.