Boundedness and stabilization in a two-species chemotaxis system with two chemicals

This paper deals with the two-species chemotaxis system with two chemicals

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\nabla\cdot(u\chi_1(v)\nabla v)+\mu_1 u(1-u-a_1w),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v-\alpha v+f_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = d_3\Delta w-\nabla\cdot(w\chi_2(z)\nabla z)+\mu_2 w(1-w-a_2u),\quad &x\in \Omega,\quad t>0,\\ z_t = d_4\Delta z-\beta z+f_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 $ ($ n\geq1 $), where the parameters $ d_1,d_2,d_3,d_4>0 $, $ \mu_1,\mu_2>0 $, $ a_1,a_2>0 $ and $ \alpha, \beta>0 $. The chemotactic function $ \chi_i $ ($ i = 1,2 $) and the signal production function $ f_i $ ($ i = 1,2 $) are smooth. If $ n = 2 $, it is shown that this system possesses a unique global bounded classical solution provided that $ |\chi'_i| $ ($ i = 1,2 $) are bounded. If $ n\leq3 $, this system possesses a unique global bounded classical solution provided that $ \mu_i $ ($ i = 1,2 $) are sufficiently large. Specifically, we first obtain an explicit formula $ \mu_{i0}>0 $ such that this system has no blow-up whenever $ \mu_i>\mu_{i0} $.

Moreover, by constructing suitable energy functions, it is shown that:

$ \bullet $ If $ a_1,a_2\in(0,1) $ and $ \mu_1 $ and $ \mu_2 $ are sufficiently large, then any global bounded solution exponentially converges to $\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$$ f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$ as $ t\rightarrow\infty $;

$ \bullet $ If $ a_1>1>a_2>0 $ and $ \mu_2 $ is sufficiently large, then any global bounded solution exponentially converges to $ (0,f_1(1)/\alpha,1,0) $ as $ t\rightarrow\infty $;

$ \bullet $ If $ a_1 = 1>a_2>0 $ and $ \mu_2 $ is sufficiently large, then any global bounded solution algebraically converges to $ (0,f_1(1)/\alpha,1,0) $ as $ t\rightarrow\infty $.