Refined description and stability for singular solutions of the 2D KellerSegel system
Abstract
We construct solutions to the two dimensional parabolicelliptic KellerSegel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $\lambda$ and of a perturbation. We rely on a detailed spectral analysis for the linearized dynamics in the parabolic neighbourhood of the singularity performed by the authors, 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 $\lambda \sim 2e^{\frac{2+\gamma}{2}}\sqrt{Tt}e^{\sqrt{\frac{\ln (Tt)}{2}}}$ where $\gamma$ is the Euler constant. This improves on the earlier result by Raphael and Schweyer 2014 and gives a new robust approach to socalled type II singularities for critical parabolic problems. A byproduct of the spectral analysis we developed is the existence of unstable blowup dynamics with speed $\lambda_\ell \sim C_0(Tt)^{\frac{\ell}{2}} \ln(Tt)^{\frac{\ell}{2(\ell  1)}}$ for $\ell \geq 2$ integer.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.00721
 Bibcode:
 2019arXiv191200721C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 67 pages. Added more comments and references and fixed typos