Global existence vs. finite-time blowup in chemotaxis models

We will discuss Keller-Segel-type models of chemotaxis, whose prominent feature is a competition between aggregating and diffusive mechanisms. Roughly, in case the aggregation prevails, short-time smooth solutions blow-up in finite time, whereas if the diffusion wins, short-time smooth solutions can be continued indefinitely. We will recall both the seminal results on the classical Keller-Segel system, and certain new ones concerning fractional and semilinear diffusions. Our focus will be the most-interesting `fair-competition regime', where the diffusion and aggregation appear to be in a balance.