Existence of Symmetric and Asymmetric Spikes for a Crime Hotspot Model
Existence of Symmetric and Asymmetric Spikes for a Crime Hotspot Model
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犯罪热点模型对称和不对称尖峰的存在性
DOI：
10.1137/130922744
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
M. Winter
中科院分区：
文献类型：
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作者：
H. Berestycki;Juncheng Wei;M. Winter
我们研究了由Short，Bertozzi和Brantingham在[SIAM J. Appl. Dyn.系统：9（2010），pp. 462- 483]。这项工作的目的是严格确定在这一模式中代表犯罪活动集中的热点的形成。更确切地说，对于一维系统，我们严格证明了以下类型的多个尖峰的稳定状态的存在性：（i）具有相同幅度的任意数量的多个尖峰（对称尖峰），以及（ii）对于一个大尖峰和一个小尖峰的情况下具有不同幅度的多个尖峰（非对称尖峰）。我们使用基于李雅普诺夫-施密特约简的方法，并将其扩展到准线性犯罪热点模型。一些新的结果，使我们能够进行李雅普诺夫-施密特约化是（i）近似的准线性犯罪热点系统的大规模的半线性Schnakenberg模型，和（ii）估计的空间依赖的第二个组件的小规模，这是一个新的方法。
We study a crime hotspot model suggested by Short, Bertozzi, and Brantingham in [SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 462--483]. The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity. More precisely, for the one-dimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: (i) multiple spikes of arbitrary number having the same amplitude (symmetric spikes), and (ii) multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). We use an approach based on Lyapunov--Schmidt reduction and extend it to the quasilinear crime hotspot model. Some novel results that allow us to carry out the Lyapunov--Schmidt reduction are (i) approximation of the quasilinear crime hotspot system on the large scale by the semilinear Schnakenberg model, and (ii) estimate of the spatial dependence of the second component on the small scale which i...