Asymptotic stabilization in a three-dimensional Keller-Segel-Stokes system with indirect signal production and subquadratic degradation

This paper deals with the Keller-Segel-Stokes system with indirect signal production and subquadratic degradation(⋆){nt+u⋅∇n=Δn−χ∇⋅(n∇v)+rn−μnα,vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut=Δu−∇P+n∇ϕ,∇⋅u=0 in a bounded domain Ω⊂R3 with smooth boundary, where χ>0, r∈R, μ>0, α∈(1,2) and ϕ∈W2,∞(Ω). A recent literature has asserted that for all reasonably regular initial data, the no-flux/no-flux/no-flux/Dirichlet initial-boundary value problem of (⋆) possesses a global bounded classical solution whenever α∈(53,2), but qualitative information on the behavior of solution for such subquadratic degradation cases has never been touched so far. The present study reveals that for the cases r>0 and r=0, such solution exponentially and algebraically approaches the trivial steady state as time goes to infinity, respectively. As for the involved case r>0, it is shown that this solution not only uniformly converges to the spatially homogeneous steady state ((rμ)1α−1,(rμ)1α−1,(rμ)1α−1,0) under an explicit largeness condition on μ, but also exponentially stabilizes toward the corresponding spatially homogeneous steady state under an implicit largeness condition on μ. Our results inter alia provide a more in-depth understanding on the asymptotic behavior of solution to system (⋆), especially explicitly determining the convergence rates for the essentially complicated case of subquadratic degradation.