Global classical solvability and asymptotic behaviours of a parabolic-elliptic chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logrithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviours of solutions in large time scales are considered as well.