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

2026/05/13 by Eliseo Luongo, Umberto Pappalettera
#math.AP #math.SP

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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(ℝⁿ) for dimensions n ∈ \3,…,9\ and throughout the supercritical range q∈ [1,(n)/(2)). An analog non-uniqueness result is given in the critical space of bounded functions taking values in L^n/2,∞(\Rⁿ). 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.

Citations

• On explosions of solutions to a system of partial differential equations modelling chemotaxis
• Quasilinear Dirichlet problems driven by positive sources

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