Abstract:

As a simplified version of a three-component chemotaxis system introduced by Jones et al to model the spatio-temporal behavior of criminal activities under the effects of police deployment, a two-component non-local system with singular sensitivity is considered over a bounded domain $\Omega\subset \mathbb{R}^n$ with $n\geq3$. For all reasonably regular initial data, the existence of classical solution of the corresponding initial-boundary value problem is established globally in time. In particular, we enlarge the range of chemotactic sensitivity $\chi_1$, compared to known results on the Short et al model which describes the spatio-temporal behavior of criminal activities without the effects of police deployment, which reveals that the police deployment has a regularization effect on solution. This is also a theoretical supplement to the previously numerical result unveiled by Jones, Brantingham and Chayes (Math Models MethodsAppl Sci, 2010) that the police deployment is beneficial for suppressing the formation of criminal hotspots.

Key words: chemotaxis system, singular sensitivity, classical solution, global existence, long-time behavior