  In the last twenty years, there have been significant advances in the study
of the blow-up phenomenon for the critical generalized Korteweg-de Vries
equation, including the determination of sufficient conditions for blowup, the
stability of blowup in a refined topology and the classification of minimal
mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and
multi-point blow-up solutions were also constructed. However, all these
results, as well as numerical simulations, involve the bubbling of a solitary
wave going at infinity at the blow-up time, which means that the blow-up
dynamics and the residue are eventually uncoupled. Even at the formal level,
there was no indication whether blowup at a finite point could occur for this
equation. In this article, we answer this question by constructing solutions
that blow up in finite time under the form of a single-bubble concentrating the
ground state at a finite point with an unforeseen blow-up rate. Finding a
blow-up rate intermediate between the self-similar rate and other rates
previously known also reopens the question of which blow-up rates are actually
possible for this equation.
