On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

This paper deals with the following two-species chemotaxis system with nonlinear diffusion, sensitivity, signal secretion and (without or with) logistic source\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \nabla \cdot (D_1(u)\nabla u - S_1(u)\nabla v) + f_{1}(u),\quad &x\in\Omega,\quad t>0,\\ v_t = \Delta v-v+g_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = \nabla \cdot (D_2(w)\nabla w - S_2(w)\nabla z) + f_{2}(w),\quad &x\in \Omega,\quad t>0,\\ z_t = \Delta z-z+g_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*} $\end{document}under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} with \begin{document}$ n\geq2 $\end{document} . The diffusion functions \begin{document}$ D_{i}(s) \in C^{2}([0,\infty)) $\end{document} and the chemotactic sensitivity functions \begin{document}$ S_{i}(s) \in C^{2}([0,\infty)) $\end{document} are given by\begin{document}$ \begin{equation*} \begin{split} D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} \quad \text{and} \quad 0 < S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} \text{ for all } s\geq0, \end{split} \end{equation*} $\end{document}where \begin{document}$ C_{d_{i}},C_{s_{i}}>0 $\end{document} and \begin{document}$ \alpha_i,\beta_{i} \in \mathbb{R} $\end{document}\begin{document}$ (i = 1,2) $\end{document} . The logistic source functions \begin{document}$ f_{i}(s) \in C^{0}([0,\infty)) $\end{document} and the nonlinear signal secretion functions \begin{document}$ g_{i}(s) \in C^{1}([0,\infty)) $\end{document} are given by\begin{document}$ \begin{equation*} \begin{split} f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} \quad \text{and} \quad g_{i}(s)\leq s^{\gamma_{i}} \text{ for all } s\geq0, \end{split} \end{equation*} $\end{document}where \begin{document}$ r_{i} \in \mathbb{R} $\end{document} , \begin{document}$ \mu_{i},\gamma_{i} > 0 $\end{document} and \begin{document}$ k_{i} > 1 $\end{document}\begin{document}$ (i = 1,2) $\end{document} . With the assumption of proper initial data regularity, the global boundedness of solution is established under the some specific conditions with or without the logistic functions \begin{document}$ f_{i}(s) $\end{document} . Moreover, in case \begin{document}$ r_{i}>0 $\end{document} , for the large time behavior of the smooth bounded solution, by constructing the appropriate energy functions, under the conditions \begin{document}$ \mu_{i} $\end{document} are sufficiently large, it is shown that the global bounded solution exponentially converges to \begin{document}$ \left((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{1}}{k_{2}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{2}}{k_{1}-1}}\right) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document} .