TY - JOUR
T1 - Boundedness and Asymptotic Behavior in a 3D Keller-Segel-Stokes System Modeling Coral Fertilization with Nonlinear Diffusion and Rotation
AU - Dai , Feng
AU - Liu , Bin
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 515
EP - 563
PY - 2022
DA - 2022/08
SN - 3
DO - http://doi.org/10.4208/csiam-am.SO-2021-0041
UR - https://global-sci.org/intro/article_detail/csiam-am/20971.html
KW - Keller-Segel-Stokes, nonlinear diffusion, tensor-valued sensitivity, boundedness, asymptotic behavior.
AB - <p style="text-align: justify;">This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization $$\begin{cases} n_t+u\cdot \nabla n=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot \nabla c)-n\rho, \\ c_t+u\cdot \nabla c=\Delta c-c+\rho, \\\rho_t+u\cdot \nabla\rho=\Delta\rho-n\rho, \\ u_t+\nabla P=\Delta u+(n+\rho)\nabla\phi, ~\nabla\cdot u=0\end{cases}$$ 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 $m&gt;0,$ $\phi∈W^{2,∞}(Ω),$ 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 $m&gt; 1−α$ for $0≤α≤\frac{2}{3},$ or $m≥ \frac{1}{3}$ for $α&gt;\frac{2}{3},$ 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.</p>