Orateur
Description
he parabolic-elliptic Keller-Segel system models cell motion under chemotaxis. It is a mass preserving equation that has the same scaling invariance as the quadratic semilinear heat equation. It admits blowup solutions in the mass critical two-dimensional case as well as in higher dimensions that are mass supercritical. In such instances when the density becomes singular in finite time, this describes cell aggregation. This talk will first review four previously known blow-up patterns (self-similar, flat, collapsing steady state, collapsing sphere). It will then present a new one in the mass critical case: where two stationary states are simultaneously collapsing and colliding at a single singular point. A formal blow-up law was proposed by Herrero-Seki-Velazquez in 2014. We provide a rigorous construction of such solution. We will explain some of the new ideas to study this dynamics that to our knowledge had not been studied before in evolution pdes, where two solitons interact in the same parabolic neighborhood from the singularity in a non-radial configuration, together with the radiation remainder. This is joint work with T.-E. Ghoul (NYU Abu Dhabi), N. Masmoudi (NYU Abu Dhabi and Courant Institute) and V. T. Nguyen (National Taiwan University).