Large time behavior in a predator-prey system with pursuit-evasion interaction

This work considers a pursuit-evasion model\begin{document}$\begin{equation} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document} with positive parameters \begin{document}$ \chi $\end{document} , \begin{document}$ \xi $\end{document} , \begin{document}$ \mu $\end{document} , \begin{document}$ \lambda $\end{document} , \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} ( \begin{document}$ N $\end{document} is the dimension of the space) with smooth boundary. We prove that if \begin{document}$ a<2 $\end{document} and \begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\} $\end{document} , (1) possesses a global bounded classical solution with a positive constant \begin{document}$ C_{\frac{N}{2}+1} $\end{document} corresponding to the maximal Sobolev regularity. Moreover, it is shown that if \begin{document}$ b\mu<\lambda $\end{document} , the solution ( \begin{document}$ u,v,w,z $\end{document} ) converges to a spatially homogeneous coexistence state with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} in the large time limit under some exact smallness conditions on \begin{document}$ \chi $\end{document} and \begin{document}$ \xi $\end{document} . If \begin{document}$ b\mu>\lambda $\end{document} , the solution converges to ( \begin{document}$ \mu,0,0,\mu $\end{document} ) with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} as \begin{document}$ t\rightarrow \infty $\end{document} under some smallness assumption on \begin{document}$ \chi $\end{document} with arbitrary \begin{document}$ \xi $\end{document} .