A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

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

$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n $ ($ n\geq1 $), where the parameters $ \chi_i>0 $, $ \mu_i>0 $ and $ a_i>0 $ ($ i = 1, 2 $). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds:

(ⅰ) $ q_1\leq a_1; $ (ⅱ) $ q_2\leq a_2 $;

(ⅲ) $ q_1>a_1 $ and $ q_2> a_2 $ as well as $ (q_1-a_1)(q_2-a_2)<1 $,

where $ q_1: = \frac{\chi_1}{\mu_1} $ and $ q_2: = \frac{\chi_2}{\mu_2} $, which partially improves the results of Zhang et al. [53] and Tu et al. [34].

Moreover, it is proved that when $ a_1, a_2\in(0, 1) $ and $ \mu_1 $ and $ \mu_2 $ are sufficiently large, then any global bounded solution exponentially converges to $ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $ as $ t\rightarrow\infty $; When $ a_1>1>a_2>0 $ and $ \mu_2 $ is sufficiently large, then any global bounded solution exponentially converges to $ (0, 1, 1, 0) $ as $ t\rightarrow\infty $; When $ a_1 = 1>a_2>0 $ and $ \mu_2 $ is sufficiently large, then any global bounded solution algebraically converges to $ (0, 1, 1, 0) $ as $ t\rightarrow\infty $. This result improves the conditions assumed in [34] for asymptotic behavior.