Details
| Original language | English |
|---|---|
| Article number | 2350032 |
| Journal | Communications in Contemporary Mathematics |
| Early online date | 29 Jul 2023 |
| Publication status | E-pub ahead of print - 29 Jul 2023 |
| Externally published | Yes |
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
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In: Communications in Contemporary Mathematics, 29.07.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Unboundedness phenomenon in a model of urban crime
AU - Fuest, Mario
AU - Heihoff, Frederic
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
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
SN - 0219-1997
M1 - 2350032
ER -