Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System. Issue 7 (25th March 2021) - Collot, Charles; Ghoul, Tej‐Eddine; Masmoudi, Nader; Nguyen, Van Tien
Abstract : We construct solutions to the two‐dimensional parabolic‐elliptic Keller‐Segel model for chemotaxis that blow up in finite time T . The solution is decomposed as the sum of a stationary state concentrated at scale λ and of a perturbation. We rely on a detailed spectral analysis for the linearised dynamics in the parabolic neighbourhood of the singularity performed by the authors in [10], providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law λ ∼ 2 e − 2 + γ 2 T − t e − ∣ ln T − t ∣ 2 where γ is the Euler constant. This improves on the earlier result by Raphael and Schweyer and gives a new robust approach to so‐called type II singularities for critical parabolic problems. A by‐product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed λ ℓ ∼ C 0 T − t ℓ 2 ln T − t − ℓ 2 ℓ − 1 for ℓ ≥ 2 integer . © 2021 Wiley Periodicals LLC.