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.

Externe Organisation(en)

Universität Paderborn

Typ

Artikel

Journal

Communications in Contemporary Mathematics

ISSN

0219-1997

Publikationsdatum

29.07.2023

Publikationsstatus

Elektronisch veröffentlicht (E-Pub)

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)