Smoothness effects of a quadratic damping term of mixed type on a chemotaxis-type system modeling propagation of urban crime

Abstract

This paper focuses on the two-dimensional Neumann initial–boundary value problem of a chemotaxis-type system with a mixed-type quadratic damping term constituting of the product of two unknown functions. It is shown that such quadratic damping term seems sufficient to exclude the possibility of blowup in infinite time. Precisely, the first result indicates that for all reasonably regular initial data and any chemotatic sensitivity, the solution of the initial–boundary value problem is global in time within a suitable generalized framework. Meanwhile, the second result demonstrates that such generalized solution enjoys the eventual boundedness and regularity properties, i.e., it becomes bounded and smooth after some waiting time. Finally, a statement on the asymptotic stability of certain steady states is derived as a by-product.