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

Overview & Synopsis

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a sphere that shrinks to a point. The singularity is in particular of type II, non self-similar and resembles a traveling wave imploding at the origin in renormalized variables. We show the stability of this dynamics among spherically symmetric solutions, and to our knowledge, this is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.