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

Abstract

This paper is concerned with a chemotaxis system in a two-dimensional setting as follows: $ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)-\kappa uv+ru-\mu u^2+ h_1, \\ &v_t = \Delta v- v+ uv+h_2, \end{split} \right. \end{equation*} $ with the parameters $ \chi, \kappa, \mu > 0 $ and $ r\in \mathbb R $, and with the given functions $ h_1, h_2\geq0 $. 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.

Keywords

• chemotaxis system
• generalized solution
• eventual smoothness
• asymptotic behavior