Unboundedness phenomenon in a model of urban crime

verfasst von
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.

Organisationseinheit(en)
Institut für Angewandte Mathematik
Externe Organisation(en)
Universität Paderborn
Typ
Artikel
Journal
Communications in Contemporary Mathematics
Band
26
ISSN
0219-1997
Publikationsdatum
29.07.2023
Publikationsstatus
Veröffentlicht
Peer-reviewed
Ja
ASJC Scopus Sachgebiete
Mathematik (insg.), Angewandte Mathematik
Ziele für nachhaltige Entwicklung
SDG 16 – Frieden, Gerechtigkeit und starke Institutionen
Elektronische Version(en)
https://doi.org/10.48550/arXiv.2109.01016 (Zugang: Offen)
https://doi.org/10.1142/S0219199723500323 (Zugang: Geschlossen)