For the $L^2$-critical generalised KdV equation, blow-up is not possible for subcritical mass elements. A minimal mass blow-up exists, as does a description of the flow for slightly supercritical mass elements. For such initial data, a finite-time blow-up occurs with $\|u_x\|_{L^2}\sim (T-t)^{-\nu}$, where $\nu$ is the blow-up rate. An infinite point blow-up occurs for $\nu \geq \frac{1}{2}$. In a certain topology, a finite-time infinite-point stable blow-up occurs at the rate of $\nu= 1$. Outside this topology, other infinite-point blow-up rates are possible, but no stability is claimed.

I will present some general results on the finite-time blow-up of the gKdV equation. Then, we will focus on results concerning finite-time infinite-point unstable (exotic) blow-up. Previously, the blow-up rate of such solutions was limited to $\nu > \frac{11}{13}$; in my latest work, this has been improved to $\nu > \frac{1}{2}$.