This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

本文重点介绍 Short等人的简化变体。模型，最初由 Rodríguez 引入，由两个耦合的类似反应扩散方程组成——其中一个模拟罪犯密度的时空演变，另一个描述吸引力场的动态. 这种模型显然与信号产生和细胞增殖和死亡的聚集的对数 Keller-Segel 模型相当。然而，令人惊讶的是，在二维环境中，该模型与经典对数 Keller-Segel 模型共享一些基本成分，具有信号吸收而不是信号产生，这是由于其对罪犯的特殊增殖和死亡机制。准确地说，它表明对于所有合理规则的初始数据，相应的初始边值问题具有全局广义解，类似于为具有信号吸收的经典对数 Keller-Segel 系统建立的解；但是，它不同于具有信号生成的对应方的通用框架。此外，它证明了这种广义解至少最终变得有界和平滑，并且还讨论了这种解的长期渐近行为。