  On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with
vanishing boundary trace is known to lead to finite-time extinction, with a
vanishing profile selected by the initial datum. In rescaled variables, we
quantify the rate of convergence to this profile uniformly in relative error,
showing the rate is either exponentially fast (with a rate constant predicted
by the spectral gap), or algebraically slow (which is only possible in the
presence of non-integrable zero modes). In the first case, the nonlinear
dynamics are well-approximated by exponentially decaying eigenmodes up to at
least twice the gap; this refines and confirms a 1980 conjecture of Berryman
and Holland. We also improve on a result of Bonforte and Figalli, by providing
a new and simpler approach which is able to accommodate the presence of zero
modes, such as those that occur when the vanishing profile fails to be isolated
(and possibly belongs to a continuum of such profiles).
