Refined Description and Stability for Singular Solutions of the 2D Keller-Segel System

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. (Formula presented.). 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. (Formula presented.)