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arXiv · 2605.13592

Spectral instability and non-uniqueness for the Keller-Segel system

Eliseo Luongo·Umberto Pappalettera
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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 \v{S}ver\'ak for the three-dimensional Navier-Stokes equations.

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Eliseo Luongo, Umberto Pappalettera. 2026-05-13. Spectral instability and non-uniqueness for the Keller-Segel system. https://arxiv.org/abs/2605.13592

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