On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime

Abstract

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - \chi \big(\frac{u}{v} \partial_x u \big)_x - uv +B_1(x,t), \qquad & x\in \Omega, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*} which was introduced by Short et al. in [Short2008] with $\chi=2$ to describe the dynamics of urban crime In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably regular nonnegative functions $B_1$ and $B_2$, we first prove the existence of global classical solutions for any choice of $\chi>0$ and all reasonably regular nonnegative initial data. We next address the issue of determining the qualitative behavior of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$. Indeed, for arbitrary $\chi>0$ we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that $u$ must decay to zero in the long-term limit. Our final result, valid for all $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $\chi=2$, states that under the above decay assumption on $B_1$, if furthermore $B_2$ appropriately stabilizes to a nontrivial function $B_{2,\infty}$, then $(u,v)$ approaches the limit $(0,v_\infty)$, where $v_\infty$ denotes the solution of \begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in \Omega, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partial\Omega. \end{array} \right. \end{eqnarray*}