Global existence of solutions for a chemotaxis-type system arising in   crime modeling
  We consider a nonlinear, strongly coupled, parabolic system arising in the modeling of burglary in residential areas. The system is of chemotaxis-type and involves a logarithmic sensivity 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 in a square, including concrete cases involving values of the parameters which are expected to be physically relevant 
Raul Manasevich, Quoc Hung Phan, Philippe Souplet
math.AP