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)