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{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*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n $ with $ n\geq2 $. The diffusion functions $ D_{i}(s) \in C^{2}([0,\infty)) $ and the chemotactic sensitivity functions $ S_{i}(s) \in C^{2}([0,\infty)) $ are given by

$ \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*} $

where $ C_{d_{i}},C_{s_{i}}>0 $ and $ \alpha_i,\beta_{i} \in \mathbb{R} $ $ (i = 1,2) $. The logistic source functions $ f_{i}(s) \in C^{0}([0,\infty)) $ and the nonlinear signal secretion functions $ g_{i}(s) \in C^{1}([0,\infty)) $ are given by

$ \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*} $

where $ r_{i} \in \mathbb{R} $, $ \mu_{i},\gamma_{i} > 0 $ and $ k_{i} > 1 $ $ (i = 1,2) $. 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 $ f_{i}(s) $.

Moreover, in case $ r_{i}>0 $, for the large time behavior of the smooth bounded solution, by constructing the appropriate energy functions, under the conditions $ \mu_{i} $ are sufficiently large, it is shown that the global bounded solution exponentially converges to $ \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) $ as $ t\rightarrow\infty $.