Global large-data generalized solutions to a two-dimensional chemotaxis system stemming from crime modelling

Abstract

<p style='text-indent:20px;'>We consider a chemotaxis system over a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb R^2 $\end{document}</tex-math></inline-formula> of the following form <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation*} \left\{ \begin{split} &amp;u_t = \Delta u-\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)- u+ a, \\ &amp;v_t = \Delta v- v+ uv(1-v)+b, \end{split} \right. \ \ \ \ \ \ (\star)\end{equation*}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>with <inline-formula><tex-math id="M2">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ a\geq0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ b&gt;0 $\end{document}</tex-math></inline-formula>. The particular version of (<inline-formula><tex-math id="M5">\begin{document}$ \star$\end{document}</tex-math></inline-formula>) with <inline-formula><tex-math id="M6">\begin{document}$ \chi = 2 $\end{document}</tex-math></inline-formula> was proposed by Pitcher to describe the evolution of population dynamics of criminal agents. Recent results reveal that the system (<inline-formula><tex-math id="M7">\begin{document}$ \star$\end{document}</tex-math></inline-formula>) associated with Neumann boundary conditions admits a global classical solution, under appropriate smallness conditions on both the initial data and the parameter <inline-formula><tex-math id="M8">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>The present study indicates that nevertheless, for all reasonably regular initial data and any <inline-formula><tex-math id="M9">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula>, the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Furthermore, it also demonstrates that, whenever <inline-formula><tex-math id="M10">\begin{document}$ a = 0 $\end{document}</tex-math></inline-formula>, such global generalized solution becomes bounded and smooth at least eventually. In particular, it approaches the spatial equilibria at exponential rate in the large time limit.