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

\begin{document}$ \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. $\end{document}

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document}), where the parameters \begin{document}$ \chi_i>0 $\end{document}, \begin{document}$ \mu_i>0 $\end{document} and \begin{document}$ a_i>0 $\end{document} (\begin{document}$ i = 1, 2 $\end{document}). 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) \begin{document}$ q_1\leq a_1; $\end{document} (ⅱ) \begin{document}$ q_2\leq a_2 $\end{document};

(ⅲ) \begin{document}$ q_1>a_1 $\end{document} and \begin{document}$ q_2> a_2 $\end{document} as well as \begin{document}$ (q_1-a_1)(q_2-a_2)<1 $\end{document},

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

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