## Details

Original language | English |
---|---|

Article number | 2350032 |

Journal | Communications in Contemporary Mathematics |

Volume | 26 |

Issue number | 7 |

Publication status | Published - 29 Jul 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.

## Keywords

- Chemotaxis, logarithmic sensitivity, singular limit, urban crime

## ASJC Scopus subject areas

**Mathematics(all)**- Mathematics(all)
**Applied Mathematics**

## Sustainable Development Goals

## Cite this

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**Unboundedness phenomenon in a model of urban crime.**/ Fuest, Mario; Heihoff, Frederic.

In: Communications in Contemporary Mathematics, Vol. 26, No. 7, 2350032, 29.07.2023.

Research output: Contribution to journal › Article › Research › peer review

*Communications in Contemporary Mathematics*, vol. 26, no. 7, 2350032. https://doi.org/10.48550/arXiv.2109.01016, https://doi.org/10.1142/S0219199723500323

*Communications in Contemporary Mathematics*,

*26*(7), Article 2350032. https://doi.org/10.48550/arXiv.2109.01016, https://doi.org/10.1142/S0219199723500323

}

TY - JOUR

T1 - Unboundedness phenomenon in a model of urban crime

AU - Fuest, Mario

AU - Heihoff, Frederic

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

PY - 2023/7/29

Y1 - 2023/7/29

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.

AB - 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.

KW - Chemotaxis

KW - logarithmic sensitivity

KW - singular limit

KW - urban crime

UR - http://www.scopus.com/inward/record.url?scp=85168005846&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2109.01016

DO - 10.48550/arXiv.2109.01016

M3 - Article

AN - SCOPUS:85168005846

VL - 26

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 7

M1 - 2350032

ER -