Unboundedness phenomenon in a model of urban crime

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Mario Fuest
  • Frederic Heihoff

Organisationseinheiten

Externe Organisationen

  • Universität Paderborn
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Details

OriginalspracheEnglisch
Aufsatznummer2350032
FachzeitschriftCommunications in Contemporary Mathematics
Jahrgang26
Ausgabenummer7
PublikationsstatusVeröffentlicht - 29 Juli 2023

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.

ASJC Scopus Sachgebiete

Ziele für nachhaltige Entwicklung

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Unboundedness phenomenon in a model of urban crime. / Fuest, Mario; Heihoff, Frederic.
in: Communications in Contemporary Mathematics, Jahrgang 26, Nr. 7, 2350032, 29.07.2023.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Fuest M, Heihoff F. Unboundedness phenomenon in a model of urban crime. Communications in Contemporary Mathematics. 2023 Jul 29;26(7):2350032. doi: 10.48550/arXiv.2109.01016, 10.1142/S0219199723500323
Fuest, Mario ; Heihoff, Frederic. / Unboundedness phenomenon in a model of urban crime. in: Communications in Contemporary Mathematics. 2023 ; Jahrgang 26, Nr. 7.
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AU - Fuest, Mario

AU - Heihoff, Frederic

N1 - Publisher Copyright: © 2023 World Scientific Publishing Company.

PY - 2023/7/29

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N2 - 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.

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