** Federico Buseghin (CYU)**

** Finite-time blow-up for the Keller–Segel system. **

The Keller–Segel system is a classical model for chemotactic aggregation and can be viewed as a nonlocal nonlinear reaction–diffusion equation.

In this talk I will discuss the existence of solutions exhibiting type II finite-time blow-up. I will first present a construction of such solutions in the two-dimensional case using gluing techniques. This construction then serves as a key ingredient in the analysis of the three-dimensional axially symmetric problem, where we obtain solutions whose mass concentrates along rings. The approach relies on an inner–outer gluing scheme adapted to the nonlocal structure of the equation, together with a refined analysis of the associated Newtonian potential near the concentration region.

This is joint work with Juan Dávila, Manuel del Pino, and Monica Musso.