Unboundedness phenomenon in a model of urban crime

Abstract

We show that spatial patterns ("hotspots") may form in the crime model \begin{equation} \left\{\; \begin{aligned} u_{t} &= \tfrac{1}{\varepsilon}\Delta u - \tfrac{\chi}{\varepsilon} \nabla \cdot \left(\tfrac{u}{v} \nabla v \right) - \varepsilon uv, \\ v_{t} &= \Delta v - v + u v, \end{aligned} \right. \end{equation} which we consider in $\Omega = B_R(0) \subset \mathbb R^n$, $R > 0$, $n \geq 3$ with $\varepsilon > 0$, $\chi > 0$ and initial data $u_0$, $v_0$ with sufficiently large initial mass $m := \int_\Omega u_0$. More precisely, for each $T > 0$ and fixed $\Omega$, $\chi$ and (large) $m$, we construct initial data $v_0$ exhibiting the following unboundedness phenomenon: Given any $M>0$, we can find $\varepsilon > 0$ such that the first component of the associated maximal solution becomes larger than $M$ at some point in $\Omega$ before the time $T$. Since the $L^1$ norm of $u$ is decreasing, this implies that some heterogeneous structure must form. We do this by first constructing classical solutions to the nonlocal scalar problem \[ w_t = \Delta w + m \frac{w^{\chi+1}}{\int_\Omega w^\chi} \] from the solutions to the crime model by taking the limit $\varepsilon \searrow 0$ under the assumption that the unboundedness phenomenon explicitly does not occur on some interval $(0,T)$. We then construct initial data for this scalar problem leading to blow-up before time $T$. As solutions to the scalar problem are unique, this proves our central result by contradiction.