Details
| Original language | English |
|---|---|
| Article number | 89 |
| Journal | Calculus of Variations and Partial Differential Equations |
| Volume | 64 |
| Issue number | 3 |
| Early online date | 17 Feb 2025 |
| Publication status | E-pub ahead of print - 17 Feb 2025 |
Abstract
We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model (Figure presented.) in a ball Ω⊂Rn with n≥2. Previous results show that unbounded solutions exist for all m,q∈R with m-q<n-2n, which, however, are necessarily global in time if q≤0. It is expected that finite-time blow-up is possible whenever q>0 but in the fully parabolic setting this has so far only been shown when max{m,q}≥1. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that (⋆) admits solutions blowing up in finite time if (Formula presented.) that is, also for certain m, q with max{m,q}<1. As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Calculus of Variations and Partial Differential Equations, Vol. 64, No. 3, 89, 17.02.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Finite-time blow-up in fully parabolic quasilinear Keller–Segel systems with supercritical exponents
AU - Cao, Xinru
AU - Fuest, Mario
N1 - Publisher Copyright: © The Author(s) 2025.
PY - 2025/2/17
Y1 - 2025/2/17
N2 - We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model (Figure presented.) in a ball Ω⊂Rn with n≥2. Previous results show that unbounded solutions exist for all m,q∈R with m-q0 but in the fully parabolic setting this has so far only been shown when max{m,q}≥1. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that (⋆) admits solutions blowing up in finite time if (Formula presented.) that is, also for certain m, q with max{m,q}<1. As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.
AB - We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model (Figure presented.) in a ball Ω⊂Rn with n≥2. Previous results show that unbounded solutions exist for all m,q∈R with m-q0 but in the fully parabolic setting this has so far only been shown when max{m,q}≥1. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that (⋆) admits solutions blowing up in finite time if (Formula presented.) that is, also for certain m, q with max{m,q}<1. As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.
UR - http://www.scopus.com/inward/record.url?scp=85219716649&partnerID=8YFLogxK
U2 - 10.1007/s00526-025-02944-4
DO - 10.1007/s00526-025-02944-4
M3 - Article
AN - SCOPUS:85219716649
VL - 64
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
SN - 0944-2669
IS - 3
M1 - 89
ER -