Identifying secondary critical mass levels in two- and higher-dimensional Keller-Segel systems
Abstract
The parabolic-elliptic Keller-Segel system { l ut = Δu − ∇ · (u∇ v), [1mm] 0 = Δv − μ+ u, μ:=1/|Ω| ∫Ωu, . (⋆) is considered under homogeneous Neumann boundary conditions in the ball Ω=BR(0)⊂ ℝn, where n≥ 2 and R>0. The main results identify a mass level m⋆(n,R)>0 as critical in the sense that • whenever 0<m<m⋆(n,R), the constant equilibrium of (⋆) given by u≡ v≡ m/|BR| is locally asymptotically stable with respect to the norm in L∞(BR), but that • if m>m⋆(n,R), then any solution of (⋆) emanating from nonnegative and nonconstant continuous radial initial data u0 with ∫BR u0=m and 1/|Br| ∫Br u0 ≥ m/|BR| for all r∈ (0,R) blows up in finite time. The value of m⋆(n,R) can be determined in an essentially explicit manner in dependence on principal zeroes of appropriate Bessel functions of the first kind; in particular, m⋆(2,R)≈ 14.68197 · π for all R>0. Accordingly, as a supplement to known mass-dependent blow-up dichotomies these findings reveal a genuinely secondary critical mass phenomenon for (⋆) especially also in its two-dimensional version in which no precedent of this form seems known.
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BibTeX
@misc{wang2026identifying,
title = {{Identifying secondary critical mass levels in two- and higher-dimensional Keller-Segel systems}},
author = {Yulan Wang and Michael Winkler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13559},
eprint = {2610.13559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13559},
}
RIS
TY - UNPB
TI - Identifying secondary critical mass levels in two- and higher-dimensional Keller-Segel systems
AU - Yulan Wang
AU - Michael Winkler
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13559
UR - https://arxiv.org/abs/2610.13559
AB - The parabolic-elliptic Keller-Segel system { l u_t = Δu − ∇ · (u∇ v), [1mm] 0 = Δv − μ+ u, μ:=1/|Ω| ∫_Ωu, . (⋆) is considered under homogeneous Neumann boundary conditions in the ball Ω=B_R(0)⊂ ℝ^n, where n≥ 2 and R>0. The main results identify a mass level m_⋆(n,R)>0 as critical in the sense that • whenever 0<m<m_⋆(n,R), the constant equilibrium of (⋆) given by u≡ v≡ m/|B_R| is locally asymptotically stable with respect to the norm in L^∞(B_R), but that • if m>m_⋆(n,R), then any solution of (⋆) emanating from nonnegative and nonconstant continuous radial initial data u_0 with ∫_B_R u_0=m and 1/|B_r| ∫_B_r u_0 ≥ m/|B_R| for all r∈ (0,R) blows up in finite time. The value of m_⋆(n,R) can be determined in an essentially explicit manner in dependence on principal zeroes of appropriate Bessel functions of the first kind; in particular, m_⋆(2,R)≈ 14.68197 · π for all R>0. Accordingly, as a supplement to known mass-dependent blow-up dichotomies these findings reveal a genuinely secondary critical mass phenomenon for (⋆) especially also in its two-dimensional version in which no precedent of this form seems known.
ER -