(untitled) (CSIAM Transactions on Applied Mathematics, 2022)
This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization cases n_t+u· abla n=Δ n^m- abla·(nS(x,n,c)· abla c)-nρ, c_t+u· abla c=Δ c-c+ρ, ρ_t+u· ablaρ=Δρ-nρ, u_t+ abla P=Δ u+(n+ρ) ablaφ, abla· u=0cases 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 sufficiently 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,∞)[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|{_Ω n_0−_Ωρ_0}_+ and ρ_∞ :=1|Ω|{_Ωρ_0−_Ω n0}_+. These results improve and extend previously known ones.