We wish to present in this talk a brief overview of recent developments in the analysis of the generalized Keller-Segel system:
\[
\partial_t \rho(t,x) = \Delta\rho^m(t,x) + \chi \nabla\cdot \left(\rho(t,x) \nabla\left(\dfrac{|x|^{k}}{k}*\rho(t,x)\right)\right)\, , \quad t>0\, , \quad x\in \mathbb{R}^N\, .
\]
Here $\rho(t,x)$ denotes the density of diffusive cells which are self-attracted through a mean field potential. This model has raised a lot of interest in the field of mathematical biology since it generally exhibits a dichotomy between global existence (dispersion of the cells) and finite time blow-up (aggregation of the cells). We will explain how this system possesses the structure of a gradient flow for the Wasserstein metric on the space of probability densities. Interestingly enough the energy functional is the sum of two opposite contributions, being respectively convex and concave, and homogeneous. We will show some consequences of this gradient flow interpretation, concerning finite time blow-up and long time asymptotics.
