On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions
Abstract
We study finite-time blowup for the parabolic--elliptic Keller--Segel system on ℝd in dimensions d≥11, where the problem is mass supercritical. For every integer l≥2, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state Q at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time T, u(t,r)=1/λ2(t)[Q(r/λ(t)) +ε(t,r/λ(t))], λ(t)=c(T−t)l/γ(d)(1+o(1)), where c>0 and γ(d)=1/2(d−2−√((d−2)(d−10))). The remainder ε converges to zero in local L∞ norms and in a range of high-order homogeneous Sobolev norms. Since 2l>γ(d), the concentration scale is strictly smaller than the parabolic scale √(T−t), and the resulting blowup is of type II. The l-th regime has exactly l−1 unstable radial modulation directions and is stable within a codimension-(l−1) class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of Q, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller--Segel flow.
Related
- Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3
- Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions
- A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
- Stability of the critical constant steady state of a Keller--Segel model
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
BibTeX
@misc{hou2026stability,
title = {{On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions}},
author = {Thomas Y. Hou and Xiang Qin and Peicong Song and Zirui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16467},
eprint = {2609.16467},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16467},
}
RIS
TY - UNPB
TI - On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions
AU - Thomas Y. Hou
AU - Xiang Qin
AU - Peicong Song
AU - Zirui Wang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16467
UR - https://arxiv.org/abs/2609.16467
AB - We study finite-time blowup for the parabolic--elliptic Keller--Segel system on ℝ^d in dimensions d≥11, where the problem is mass supercritical. For every integer l≥2, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state Q at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time T, u(t,r)=1/λ^2(t)[Q(r/λ(t)) +ε(t,r/λ(t))], λ(t)=c(T−t)^l/γ(d)(1+o(1)), where c>0 and γ(d)=1/2(d−2−√((d−2)(d−10))). The remainder ε converges to zero in local L^∞ norms and in a range of high-order homogeneous Sobolev norms. Since 2l>γ(d), the concentration scale is strictly smaller than the parabolic scale √(T−t), and the resulting blowup is of type II. The l-th regime has exactly l−1 unstable radial modulation directions and is stable within a codimension-(l−1) class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of Q, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller--Segel flow.
ER -