Boundedness and Asymptotic Behavior in a 3D Keller-Segel-Stokes System Modeling Coral Fertilization with Nonlinear Diffusion and Rotation

Abstract

. This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization in a bounded and smooth domain Ω ⊂ R 3 with zero-flux boundary for n , c , ρ and no-slip boundary for u , where m > 0, φ ∈ W 2, ∞ ( Ω ) , and S : ¯ Ω × [ 0, ∞ ) 2 → R 3 × 3 is given suffi-ciently smooth function such that | S ( x , n , c ) |≤ S 0 ( c )( n + 1 ) − α for all ( x , n , c ) ∈ ¯ Ω × [ 0, ∞ ) 2 with α ≥ 0 and some nondecreasing function S 0 : [ 0, ∞ ) 7→ [ 0, ∞ ) . It is shown that if m > 1 − α for 0 ≤ α ≤ 23 , or m ≥ 13 for α > 23 , then for any reasonably regular initial data, the corresponding initial-boundary value problem admits at least one globally bounded weak solution which stabilizes to the spatially homogeneous equilibrium ( n ∞ , ρ ∞ , ρ ∞ ,0 ) in an appropriate sense, where n ∞ : = 1 | Ω | (cid:8) R Ω n 0 − R Ω ρ 0 (cid:9) + and ρ ∞ : = 1 | Ω | (cid:8) R Ω ρ 0 − R Ω n 0 (cid:9) + . These results improve and extend previously known ones.