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• W3207088029 abstract "<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}$begin{equation} left{ begin{split} &amp;u_t = Delta u-chinablacdot(unabla w)+u(mu-u+av), &amp;v_t = Delta v+xinablacdot(vnabla z)+v(lambda-v-bu), &amp;w_t = Delta w-w+v, &amp;z_t = Delta z-z+u end{split} right. (1) end{equation}$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 $end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id=M7>begin{document}$ Omegasubsetmathbb{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} $end{document}</tex-math></inline-formula> corresponding to the maximal Sobolev regularity. Moreover, it is shown that if <inline-formula><tex-math id=M12>begin{document}$ bmu&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 $end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id=M17>begin{document}$ bmu&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) $end{document}</tex-math></inline-formula> as <inline-formula><tex-math id=M20>begin{document}$ trightarrow 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>" @default.
• W3207088029 created "2021-10-25" @default.
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• W3207088029 date "2022-01-01" @default.
• W3207088029 modified "2023-09-26" @default.
• W3207088029 title "Large time behavior in a predator-prey system with pursuit-evasion interaction" @default.
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• W3207088029 doi "https://doi.org/10.3934/dcdsb.2021240" @default.
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