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 {n(t) + u center dot del n = Delta n -chi del center dot (n del v)+ rn- mu n(alpha), v(t) + u center dot del v = Delta v - v + w, w(t) + u center dot del w = Delta w - w + n, ut = Delta u-del P + n del phi, del center dot u = 0 ( *) in a bounded domain Omega subset of R-3 with smooth boundary, where chi > 0, r is an element of R, mu > 0, alpha is an element of(1, 2) and phi is an element of W-2,W-infinity (Omega). A recent literature has asserted that for all reasonably regular initial data, the no-flux/no-flux/noflux/Dirichlet initial-boundary value problem of ( *) possesses a global bounded classical solution whenever alpha is an element of( 5/3, 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> 0and 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 spatiallyKeller-Segel-Stokes system; Indirect signal production; Subquadratic degradation; Asymptotic stabilization homogeneous steady state (( r/mu) (1/alpha- 1), ( r/mu) (1/alpha- 1), ( r/mu) (1/alpha- 1), 0) under an explicit largeness condition on mu, but also exponentially stabilizes toward the corresponding spatially homogeneous steady state under an implicit largeness condition on mu. 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. (c) 2023 Elsevier Inc. All rights reserved.