Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term

TL;DR

This paper constructs generalized solutions for a criminology-inspired PDE system with logistic growth and nonlinear interactions, demonstrating conditions under which classical solutions exist and analyzing the system's behavior.

Contribution

It introduces a framework for generalized solutions to a novel PDE system modeling crime hot spots, incorporating logistic terms and nonlinear interactions, extending prior models.

Findings

Existence of generalized solutions for the system.

Strengthening the logistic term yields classical solutions.

Analysis of the interplay between nonlinear terms.

Abstract

We consider the system \[ \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot ( \tfrac{u}{v} \nabla v) - uv + \rho u - \mu u^2, \\ v_t &= \Delta v - v + u v \end{aligned} \right. \tag{$\star$} \] with $\rho \in \mathbb{R}, \mu > 0, \chi > 0$ in a bounded domain $\Omega \subseteq \mathbb{R}^2$ with smooth boundary. While very similar to chemotaxis models from biology, this system is in fact inspired by recent modeling approaches in criminology to analyze the formation of crime hot spots in cities. The key addition here in comparison to similar models is the logistic source term. The central complication this system then presents us with, apart from us allowing for arbitrary $\chi > 0$, is the nonlinear growth term $uv$ in the second equation as it makes obtaining a priori information for $v$ rather difficult. Fortunately, it is somewhat tempered by its negative counterpart and the logistic source term in the first equation. It is this interplay that still gives us enough access to a priori information to achieve the main result of this paper, namely the construction of certain generalized solutions to ($\star$). To illustrate how close the interaction of the $uv$ term in the second equation and the $-\mu u^2$ term in the first equation is to granting us classical global solvability, we further give a short argument showing that strengthening the $-\mu u^2$ term to $-\mu u^{2+\gamma}$ with $\gamma > 0$ in the first equation directly leads to global classical solutions.