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

Zixuan Qiu; Bin Li; Zixuan Qiu; Bin Li

doi:10.3934/era.2023163

Electronic Research Archive

2023, Volume 31, Issue 6: 3218-3244. doi: 10.3934/era.2023163

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Research article

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

Zixuan Qiu ,
Bin Li ^,

School of Science, Ningbo University of Technology, Ningbo 315211, China

Academic Editor: Alain Miranville

Received: 08 February 2023 Revised: 15 March 2023 Accepted: 20 March 2023 Published: 28 March 2023

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.
- chemotaxis system,
- generalized solution,
- eventual smoothness,
- asymptotic behavior
Citation: Zixuan Qiu, Bin Li. Eventual smoothness of generalized solutions to a singular chemotaxis system for urban crime in space dimension 2[J]. Electronic Research Archive, 2023, 31(6): 3218-3244. doi: 10.3934/era.2023163

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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.

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