Axisymmetric type II blowup solutions to the three-dimensional Keller-Segel system

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blow up in finite time.

In particular, the singularity is of type II, which locally admits a leading-order profile of the rescaled stationary solution of the two-dimensional system.

Additionally, mass concentration occurs along a one-dimensional ring in the plane.

In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.