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

<p style='text-indent:20px;'>This work considers a pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1000">\begin{document}$\begin{equation} \left\{ \begin{split} &amp;u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &amp;v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &amp;w_t = \Delta w-w+v,\\ &amp;z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>with positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula> is the dimension of the space) with smooth boundary. We prove that if <inline-formula><tex-math id="M9">\begin{document}$ a&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}&gt;\max\{\chi,\xi\} $\end{document}</tex-math></inline-formula>, (1) possesses a global bounded classical solution with a positive constant <inline-formula><tex-math id="M11">\begin{document}$ C_{\frac{N}{2}+1} $\end{document}</tex-math></inline-formula> corresponding to the maximal Sobolev regularity. Moreover, it is shown that if <inline-formula><tex-math id="M12">\begin{document}$ b\mu&lt;\lambda $\end{document}</tex-math></inline-formula>, the solution (<inline-formula><tex-math id="M13">\begin{document}$ u,v,w,z $\end{document}</tex-math></inline-formula>) converges to a spatially homogeneous coexistence state with respect to the norm in <inline-formula><tex-math id="M14">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> in the large time limit under some exact smallness conditions on <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M17">\begin{document}$ b\mu&gt;\lambda $\end{document}</tex-math></inline-formula>, the solution converges to (<inline-formula><tex-math id="M18">\begin{document}$ \mu,0,0,\mu $\end{document}</tex-math></inline-formula>) with respect to the norm in <inline-formula><tex-math id="M19">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M20">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula> under some smallness assumption on <inline-formula><tex-math id="M21">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> with arbitrary <inline-formula><tex-math id="M22">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>.</p>