Details
| Originalsprache | Englisch |
|---|---|
| Aufsatznummer | 108788 |
| Fachzeitschrift | Applied mathematics letters |
| Jahrgang | 146 |
| Frühes Online-Datum | 19 Juli 2023 |
| Publikationsstatus | Veröffentlicht - Dez. 2023 |
Abstract
Our main result shows that the mass 2π is critical for the minimal Keller–Segel system ut=Δu−∇⋅(u∇v),vt=Δv−v+u,considered in a quarter disc Ω={(x1,x2)∈R2∣x1>0,x2>0,x12+x22<R2}, R>0, in the following sense: For all reasonably smooth nonnegative initial data u0,v0 with ∫Ωu0<2π, there exists a global classical solution to the Neumann initial boundary value problem associated to (⋆), while for all m>2π there exist nonnegative initial data u0,v0 with ∫Ωu0=m so that the corresponding classical solution of this problem blows up in finite time. At the same time, this gives an example of boundary blow-up in (⋆). Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic–elliptic simplifications of (⋆) only.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Angewandte Mathematik
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in: Applied mathematics letters, Jahrgang 146, 108788, 12.2023.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Corners and collapse
T2 - Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller–Segel system
AU - Fuest, Mario
AU - Lankeit, Johannes
PY - 2023/12
Y1 - 2023/12
N2 - Our main result shows that the mass 2π is critical for the minimal Keller–Segel system ut=Δu−∇⋅(u∇v),vt=Δv−v+u,considered in a quarter disc Ω={(x1,x2)∈R2∣x1>0,x2>0,x12+x22<R2}, R>0, in the following sense: For all reasonably smooth nonnegative initial data u0,v0 with ∫Ωu0<2π, there exists a global classical solution to the Neumann initial boundary value problem associated to (⋆), while for all m>2π there exist nonnegative initial data u0,v0 with ∫Ωu0=m so that the corresponding classical solution of this problem blows up in finite time. At the same time, this gives an example of boundary blow-up in (⋆). Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic–elliptic simplifications of (⋆) only.
AB - Our main result shows that the mass 2π is critical for the minimal Keller–Segel system ut=Δu−∇⋅(u∇v),vt=Δv−v+u,considered in a quarter disc Ω={(x1,x2)∈R2∣x1>0,x2>0,x12+x22<R2}, R>0, in the following sense: For all reasonably smooth nonnegative initial data u0,v0 with ∫Ωu0<2π, there exists a global classical solution to the Neumann initial boundary value problem associated to (⋆), while for all m>2π there exist nonnegative initial data u0,v0 with ∫Ωu0=m so that the corresponding classical solution of this problem blows up in finite time. At the same time, this gives an example of boundary blow-up in (⋆). Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic–elliptic simplifications of (⋆) only.
KW - Blow-up
KW - Chemotaxis
KW - Critical mass
KW - Global existence
UR - http://www.scopus.com/inward/record.url?scp=85166298066&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2305.18839
DO - 10.48550/arXiv.2305.18839
M3 - Article
AN - SCOPUS:85166298066
VL - 146
JO - Applied mathematics letters
JF - Applied mathematics letters
SN - 0893-9659
M1 - 108788
ER -