| Abstract: |
| In this talk we introduce the mathematical models representing the fertilization process of corals and consider the following full model
\begin{equation*}
\left\{
\begin{array}{ll}
\rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho S(x,\rho,c)\nabla c)-\rho e,
\
e_t+u\cdot\nabla e=\Delta e-\rho e,
\
c_t+u\cdot\nabla c=\Delta c-c+e,
\
u_t=\Delta u-\nabla P+(\rho+e)\nabla\phi,\quad \nabla\cdot u=0,
\end{array}\right.
\end{equation*}
in a bounded domain $\Omega\in\mathbb R^3$ with zero-flux boundary for $\rho$, $e$, $c$ and no-slip boundary for $u$. The chemotactic sensitivity
$S$ is not a scalar function but rather attains values in $\mathbb R^{3\times 3}$, and satisfies
$|S(x,\rho,c)|\leq C_S(1+\rho)^{-\alpha}$ with some $C_S>0$ and $\alpha>0$.
In this talk, we will show the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption $\alpha>\frac13$. Moreover, all the solutions are shown to decay to a spatially homogeneous equilibrium
exponentially as time goes to infinity. |
|