Roles of logistic source in a singular chemotaxis urban crime model

In this paper, we systematically exhibit the roles of logistic source on global boundedness and convergence of classical solutions, and global existence and eventual smoothness as well as convergence of generalized solutions for a singular chemotaxis urban crime model. Roughly speaking, this work shows that suitably (can be precise) strong logistic damping ensures boundedness and convergence of classical solutions, and weak logistic damping ensures global existence of generalized solution and its eventual smoothness as well as convergence in two and three dimensions. The former in particular implies that strong logistic damping can prevent the formation of crime hotspots (spatial patterns). Previous known results in this respect are thus significantly improved and extended.