Spectral instability and non-uniqueness for the Keller-Segel system
- Authors: Eliseo Luongo, Umberto Pappalettera
- Preprint year: 2026
- First public date: 2026-05-13
- arXiv: 2605.13592
Abstract
We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$. An analog non-uniqueness result is given in the critical space of bounded functions taking values in $L^{n/2,\infty}(\R^n)$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and Šverák for the three-dimensional Navier-Stokes equations.