Boundedness and Asymptotic Behavior in a 3D Keller-Segel-Stokes System Modeling Coral Fertilization with Nonlinear Diffusion and Rotation (CSIAM Transactions on Applied Mathematics, 2022)
This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization $$\begin{cases} n_t+u\cdot abla n=\Delta n^m- abla\cdot(nS(x,n,c)\cdot abla c)-n\rho, \\ c_t+u\cdot abla c=\Delta c-c+\rho, \\\rho_t+u\cdot abla\rho=\Delta\rho-n\rho, \\ u_t+ abla P=\Delta u+(n+\rho) abla\phi, ~ abla\cdot u=0\end{cases}$$ in a bounded and smooth domain $Ω⊂\mathbb{R}^3$ with zero-flux boundary for $n,$ $c,$ $\rho$ and no-slip boundary for $u,$ where $m>0,$ $\phi∈W^{2,∞}(Ω),$ and $S: \overline{Ω}×[0,∞)^ 2→\mathbb{R}^{3×3}$ is given sufficiently smooth function such that $|S(x,n,c)|≤S_0(c)(n+1)^{−α}$ for all $(x,n,c)∈\overline{Ω}×[0,∞)^2$ with $α≥0$ and some nondecreasing function $S_0 : [0,∞)\mapsto [0,∞).$ It is shown that if $m> 1−α$ for $0≤α≤\frac{2}{3},$ or $m≥ \frac{1}{3}$ for $α>\frac{2}{3},$ 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 $(n_∞,\rho_∞,\rho_∞,0)$ in an appropriate sense, where $n_∞ := \frac{1}{|Ω|}\{\int_Ω n_0−\int_Ω\rho_0\}_+$ and $\rho_∞ :=\frac{1}{|Ω|}\{\int_Ω\rho_0−\int_Ω n0\}_+.$ These results improve and extend previously known ones.
Keywords: Keller-Segel-Stokes; nonlinear diffusion; tensor-valued sensitivity; boundedness; asymptotic behavior.