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Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$

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arxiv 2309.13932 v2 pith:HR3QPNVA submitted 2023-09-25 math.AP

V. T. Nguyen , N. Nouaili , H. Zaag This is my paper · ORCID
classification math.AP
keywords blowupkeller-segelquadsystemdeltafracmathcalnabla
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abstract

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system $\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \textup{in}\;\; \mathbb{R}^d,\; d = 3,4,$ and derive the final blowup profile $ u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \textup{as}\;\; r \to 0, \;\; c_d > 0.$ To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999).

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Cited by 2 Pith papers

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

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

    math.AP 2025-01 conditional novelty 8.0 of 10

    Any sufficiently small H^2 nonradial perturbation of the explicit 3D Keller-Segel self-similar blowup profile still blows up along the same self-similar profile with decaying error.

  2. 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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