Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci 107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst. 9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems.

中文翻译：

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趋化性尖峰解的存在和稳定性是犯罪模式形成的系统建模

入室盗窃等城市犯罪是每个主要城市地区的社会问题。因此，已经提出了许多数学模型来研究这些犯罪的集体行为。在 [VB Pasour、GE Tita、PJ Brantingham、AL Bertozzi、MB Short、MR D'Orsogna 和 LB Chayes，犯罪行为的统计模型，数学。方法应用程序。科学 107 (2008) 1249–1267；MB Short，AL Bertozzi 和 PJ Brantingham，城市犯罪中的非线性模式：热点、分叉和抑制，SIAM J. Appl。达因。系统。9 (2010) 462–483]，肖特等人。提出了一种基于代理的住宅盗窃统计模型来模拟犯罪热点现象。从反应-扩散系统的角度来看，该模型是具有交叉扩散的趋化系统，呈现出热点现象。在本文中，我们首先构造该系统的径向热点解，然后通过研究非局部特征值问题来研究该热点解的线性稳定性。事实证明，热点的稳定性完全不同，具体取决于系统所处的空间维度。由于交叉扩散引起的准线性，该系统的主要数学困难涉及处理热点核心附近扩散的急剧变化。我们相信本文中使用的技术可以开发用于治疗许多其他趋化系统。由于交叉扩散引起的准线性，该系统的主要数学困难涉及处理热点核心附近扩散的急剧变化。我们相信本文中使用的技术可以开发用于治疗许多其他趋化系统。由于交叉扩散引起的准线性，该系统的主要数学困难涉及处理热点核心附近扩散的急剧变化。我们相信本文中使用的技术可以开发用于治疗许多其他趋化系统。