Blow-up patterns for the Fujita equation

The Fujita equation (a semilinear heat equation with a power type nonlinearity) is among the simplest and the most thoroughly studied non-linear parabolic PDEs. Put forward in the sixties, it has been a canonical case study of the singularity formation ever since. Despite its apparent simplicity it displays a staggering complexity of blow-up dynamics much of which remains unexplored. In particular, the description of blow-up in the so-called Sobolev supercritical regime poses a challenge that calls for new insights and methods.
In my talk I will describe the intricacies of the singularity formation for this model, outline what's known, what's fresh and what's conjectural and present some important open problems.