Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime

We consider the spatially two-dimensional version of a cross-diffusion system, as originally proposed by Short et al. in [M.B. Short, M.R. D’Orsogna, V.B. Pasour, G.E. Tita, P.J. Brantingham, A.L. Bertozzi, and L.B. Chayes, Math. Mod. Meth. Appl. Sci., 18:1249–1267, 2008] to describe the evolution of urban crime. Although sharing some basic structure elements with the well-studied classical Keller–Segel chemotaxis model, this system contains an essential difference to the latter by accounting for a certain nonlinear mechanism of attractant production, potentially yet increasing explosion-supporting properties.

The intention of this paper is to make sure that despite this, a theory of global smooth solutions can be established after all within certain small-data settings which can be described in an essentially explicit manner. The main results in this direction firstly identify hypotheses on smallness of the initial data, and of some given external production terms, as sufficient to ensure global existence and uniqueness of smooth and bounded solutions. Secondly, any such bounded solution is shown to asymptotically approach some steady state, provided that the prescribed sources comply with appropriate additional assumptions on their stabilization in the large-time limit. Finally, a statement on asymptotic stability of certain steady states is derived as a by-product.

Keywords

crime propagation, chemotaxis, global existence, large-time behavior, stability

2010 Mathematics Subject Classification

35B40, 35K55, 35Q91

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Youshan Tao was supported by the National Natural Science Foundation of China (No. 11861131003). Michael Winkler acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project “Emergence of structures and advantages in cross-diffusion systems” (No. 411007140, GZ:WI 3707/5-1).

Received 5 August 2019

Accepted 4 November 2020

Published 5 May 2021