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We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq3$, which is the mass supercritical case. This system is known to exhibit richdynamical behavior including singularity formation via self-similar solutions.An explicit example has been found more than two decades ago by Brenner et al.\cite{BCKSV99}, and is conjectured to be nonlinearly radially stable. We provethis conjecture for $d=3$. Our approach consists of reformulating the problemin similarity variables and studying the Cauchy evolution in intersectionSobolev spaces via semigroup theory methods. To solve the underlying spectralproblem, we crucially rely on a technique we recently developed in\cite{GloSch20}. To our knowledge, this provides the first result on stableself-similar blowup for the Keller-Segel system. Furthermore, the extension ofour result to any higher dimension is straightforward. We point out that ourapproach is general and robust, and can therefore be applied to a wide class ofparabolic models.

Stable singularity formation for the Keller-Segel system in three dimensions