Unboundedness phenomenon in a model of urban crime

authored by

Mario Fuest, Frederic Heihoff

Abstract

We show that spatial patterns ("hotspots") may form in the crime model ut = 1 u -χ u vv - uv,vt = v - v + uv, which we consider in ω = BR(0) n, R > 0, n ≥ 3 with > 0, χ > 0 and initial data u0, v0 with sufficiently large initial mass m:= ωu0. More precisely, for each T > 0 and fixed ω, χ and (large) m, we construct initial data v0 exhibiting the following unboundedness phenomenon: Given any M > 0, we can find > 0 such that the first component of the associated maximal solution becomes larger than M at some point in ω before the time T. Since the L1 norm of u is decreasing, this implies that some heterogeneous structure must form. We do this by first constructing classical solutions to the nonlocal scalar problem wt = w + m ωwχ-1wχ+1, from the solutions to the crime model by taking the limit 0 under the assumption that the unboundedness phenomenon explicitly does not occur on some interval (0,T). We then construct initial data for this scalar problem leading to blow-up before time T. As solutions to the scalar problem are unique, this proves our central result by contradiction.

Organisation(s)

Institute of Applied Mathematics

External Organisation(s)

Paderborn University

Type

Article

Journal

Communications in Contemporary Mathematics

Volume

26

ISSN

0219-1997

Publication date

29.07.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Mathematics(all), Applied Mathematics

Sustainable Development Goals

SDG 16 - Peace, Justice and Strong Institutions

Electronic version(s)

https://doi.org/10.48550/arXiv.2109.01016 (Access: Open)
https://doi.org/10.1142/S0219199723500323 (Access: Closed)