Abstract:

The talk will give an update on the study of singularities in the two dimensional Keller-Segel equation modeling chemotactic aggregation in biology. It is a nonlocal nonlinear reaction-diffusion equation and  -critical in the two dimensional case in the sense that the total mass of the solution is invariant under the scaling symmetry of the problem. It's well known that its solution does not exist globally if the total mass is strictly greater than a critical threshold (the solution blows up in finite time). In the radially symmetric setting, the equation can be transformed to a local similinear heat equation from which formal computations of blowup had been derived by Herrero-Velazquez, Dejak-Lushnikov-Yu-Ovchinnikov-Sigal, Dyachenko-Lushnikov-Vladimirova via matching asymptotics. A rigorous analysis in the radial case was provided later by Raphael-Schweyer replying on the so-called modulation technique. I will present a new framework based on a combination of  a spectral analysis and a robust energy-type method that allows us to tackle the full nonradial problem and obtain refined description of the blowup dynamics. Singularities formation  in higher cases (-supercritical)  and related problems will also be discussed.