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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 boundeddomain $\Omega \subseteq \mathbb{R}^2$ with smooth boundary. While very similarto chemotaxis models from biology, this system is in fact inspired by recentmodeling approaches in criminology to analyze the formation of crime hot spotsin cities. The key addition here in comparison to similar models is thelogistic source term.  The central complication this system then presents us with, apart from usallowing for arbitrary $\chi > 0$, is the nonlinear growth term $uv$ in thesecond equation as it makes obtaining a priori information for $v$ ratherdifficult. Fortunately, it is somewhat tempered by its negative counterpart andthe logistic source term in the first equation. It is this interplay that stillgives us enough access to a priori information to achieve the main result ofthis paper, namely the construction of certain generalized solutions to($\star$).  To illustrate how close the interaction of the $uv$ term in the secondequation and the $-\mu u^2$ term in the first equation is to granting usclassical global solvability, we further give a short argument showing thatstrengthening the $-\mu u^2$ term to $-\mu u^{2+\gamma}$ with $\gamma > 0$ inthe first equation directly leads to global classical solutions.

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