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Title: | Existence of symmetric and asymmetric spikes for a crime hotspot model |
Authors: | Wei, J Winter, M |
Keywords: | Crime model;Reaction-diffusion systems;Multiple spikes;Symmetric and asymmetric;Quasilinear chemotaxis system;Schnakenberg model;Lyapunov–Schmidt reduction |
Issue Date: | 2014 |
Publisher: | Society for Industrial and Applied Mathematics |
Citation: | SIAM Journal on Mathematical Analysis, 46(1), 691–719, 2014 |
Abstract: | 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 is dominated by the quasilinear part of the system. The paper concludes with an extension to the anisotropic case. |
Description: | Copyright @ 2014 Society for Industrial and Applied Mathematics |
URI: | http://epubs.siam.org/doi/abs/10.1137/130922744 http://bura.brunel.ac.uk/handle/2438/8021 |
DOI: | http://dx.doi.org/10.1137/130922744 |
ISSN: | 0036-1410 |
Appears in Collections: | Publications Dept of Mathematics Research Papers Mathematical Sciences |
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