Eventual smoothness of generalized solutions to a singular chemotaxis system for urban crime in space dimension 2

This paper is concerned with a chemotaxis system in a two-dimensional setting as follows: {u(t) = Delta u - chi del . (u del lnv) - kappa uv + ru - mu u(2) + h(1), v(t) = Delta v - v + uv + h(2), with the parameters chi, kappa, mu > 0 and r is an element of R, and with the given functions h(1), h(2) >= 0. This model was originally introduced by Short et al for urban crime with the particular values chi = 2, r = 0 and mu = 0, and the logistic source term ru - mu u(2) was incorporated into (star) by Heihoff to describe the fierce competition among criminals. Heihoff also proved that the initial-boundary value problem of (star) possesses a global generalized solution in the two-dimensional setting. The main purpose of this paper is to show that such a generalized solution becomes bounded and smooth at least eventually. In addition, the long-time asymptotic behavior of such a solution is discussed.