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Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions

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arxiv 2501.07073 v2 pith:H5Y5DPZB submitted 2025-01-13 math.AP

Zexing Li , Tao Zhou This is my paper · ORCID
classification math.AP
keywords stabilityanalysisoperatordimensionsexplicitkeller-segelmethodmode
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In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in [Brenner-Constantin-Kadanoff-Schenkel-Venkataramani, 1999], was recently confirmed by [Glogi\'c-Sch\"orkhuber, 2024]. This paper aims to extend the radial stability to the nonradial setting, building on the finite-codimensional stability analysis in our previous work [Li-Zhou, 2024]. The main input is the mode stability of the linearized operator, whose nonlocal nature presents essential challenges for the spectral analysis. Besides a quantitative perturbative analysis for the high spherical classes, we adapt in the first spherical class the wave operator method of [Li-Wei-Zhang, 2020] for the fluid stability to localize the operator and remove the known unstable mode simultaneously. Our method provides localization beyond the partial mass variable and is independent of the explicit formula of the profile, so it potentially sheds light on other linear nonlocal problems.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

    math.AP 2025-07 accept novelty 7.0 of 10

    For slightly mass-supercritical NLS in any dimension, the low-energy unstable spectrum of the self-similar linearized operator consists exactly of the symmetry modes 0, -bi, and -2bi.

  2. Finite time blow-up for an inhomogeneous parabolic equation

    math.AP 2026-07 accept novelty 6.0 of 10

    For large n, a codimension-n Lipschitz manifold of nonradial data produces finite-time blow-up to the homogeneous self-similar profile Φ_n for the inhomogeneous heat equation in R^3 with p>5.

  3. Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system

    math.AP 2025-08 conditional novelty 6.0 of 10

    For any finite set of points in the half-plane of axial symmetry, there exists a 3D Keller-Segel solution whose mass concentrates at those points with a precisely quantified finite-time blow-up rate.

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