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:

$$\{\begin{array}{rl}& u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot \left(u\mathrm{\nabla }\ln v\right)-\kappa uv+ru-\mu u^2+h_1,\\ & v_t=\mathrm{\Delta }v-v+uv+h_2,\end{array}(\star )$$

with the parameters $\chi ,\kappa ,\mu >0$ and $r\in \mathbb{R}$, and with the given functions $h_1,h_2\ge 0$. 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.