Global existence of solutions for a chemotaxis-type system arising in crime modelling

Abstract

We consider a nonlinear, strongly coupled, parabolic system arising in the modelling of burglary in residential areas. This model appeared in Pitcher (Eur. J. Appl. Math., 2010, Vol. 21, pp. 401–419), as a more realistic version of the Short et al. (Math. Models Methods Appl. Sci., 2008, Vol. 18, pp. 1249–1267) model. The system under consideration is of chemotaxis-type and involves a logarithmic sensitivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain is a square, including concrete cases involving values of the parameters which are expected to be physically relevant.