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We consider the parabolic-elliptic Keller-Segel system in three dimensionsand higher, corresponding to the mass supercritical case. We constructrigorously a solution which blows up in finite time by having its massconcentrating near a ring that shrinks to a point. In particular, thesingularity is of type II, non self-similar. We show the stability of thisdynamics among spherically symmetric solutions. In renormalised variables, thesolution ressembles a traveling wave imploding at the origin, and this, to ourknowledge, is the first stability result for such phenomenon for an evolutionPDE. We develop a framework to handle the interactions between the two blowupzones contributing to the mechanism: a thin inner zone around the ring whereviscosity effects occur, and an outer zone where the evolution is mostlyinviscid.

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher