Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production

This paper deals with the following quasilinear two-species chemotaxis system \begin{document}$ \begin{equation*} \begin{cases} \partial_{t} u_1 = \nabla \cdot (D_1(u_{1})\nabla u_{1} - S_1(u_{1})\nabla v) + f_{1}(u_{1}),\quad x'>under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^{n} $\end{document} \begin{document}$ (n\geq2) $\end{document} . The diffusivity and the density-dependent sensitivity are given by \begin{document}$ D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} $\end{document} and \begin{document}$ S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} $\end{document} for all \begin{document}$ s\geq0 $\end{document} , respectively, where \begin{document}$ C_{d_{i}},C_{s_{i}}>0 $\end{document} and \begin{document}$ \alpha_i,\beta_{i} \in \mathbb{R} $\end{document} ; the logistic source and the signal productions are given by \begin{document}$ f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} $\end{document} and \begin{document}$ g_{i}(s)\leq s^{\gamma_{i}} $\end{document} for all \begin{document}$ s\geq0 $\end{document} respectively, where \begin{document}$ 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) $\end{document} . It is proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions \begin{document}$ f_{i}(s) $\end{document} , which partially improves the results in [ 25 ]. Moreover, in case \begin{document}$ r_{i}>0 $\end{document} , if \begin{document}$ \mu_{i} $\end{document} are sufficiently large, it is shown that the global bounded solution exponentially converges to \begin{document}$ ((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{1}}{k_{1}-1}} + (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{2}}{k_{2}-1}}) $\end{document} as \begin{document}$ t\rightarrow \infty $\end{document} .