Boundedness and asymptotic behavior to a chemotaxis-fluid system with singular sensitivity and logistic source

Abstract

In this paper, we consider the following chemotaxis-fluid model with singular sensitivity and logistic source{nt+u⋅∇n=Δn−χ∇⋅(nc∇c)+rn−μnk,x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+λ(u⋅∇)u=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 in a bounded domain Ω⊂RN(N=2,3) with smooth boundary ∂Ω. Under the non-flux boundary conditions for n and c, and the non-slip boundary condition for u, we establish the global boundedness and the time-decay rates of the classical solutions for any k>1 provided that χ satisfies suitable restrictions.