Unboundedness phenomenon in a model of urban crime

TL;DR

This paper demonstrates that in a specific urban crime model, spatial hotspots can form with unbounded intensity, showing that solutions can blow up in finite time under certain initial conditions, indicating pattern formation.

Contribution

The authors construct initial data leading to solution blow-up in a crime model, revealing unboundedness phenomena and pattern formation in the system.

Findings

Solutions can become arbitrarily large before time T.

Unboundedness implies formation of heterogeneous crime hotspots.

Constructed initial data causes finite-time blow-up.

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.