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{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\in\Omega,\quad t>0,\\ \partial_{t} u_2 = \nabla \cdot (D_2(u_{2})\nabla u_{2} - S_2(u_{2})\nabla v) + f_{2}(u_{2}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} v = \Delta v-v+g_1(u_{1})+g_2(u_{2}),\quad &x\in\Omega,\quad t>0 \end{cases} \end{equation*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^{n} $ $ (n\geq2) $. The diffusivity and the density-dependent sensitivity are given by $ D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} $ and $ S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} $ for all $ s\geq0 $, respectively, where $ C_{d_{i}},C_{s_{i}}>0 $ and $ \alpha_i,\beta_{i} \in \mathbb{R} $; the logistic source and the signal productions are given by $ f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} $ and $ g_{i}(s)\leq s^{\gamma_{i}} $ for all $ s\geq0 $ respectively, where $ r_{i} \in \mathbb{R} $, $ \mu_{i},\gamma_{i} > 0 $ and $ k_{i} > 1 $ $ (i = 1,2) $. It is proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions $ f_{i}(s) $, which partially improves the results in [25]. Moreover, in case $ r_{i}>0 $, if $ \mu_{i} $ are sufficiently large, it is shown that the global bounded solution exponentially converges to $ ((\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}}) $ as $ t\rightarrow \infty $.