  We study the extinction behavior of solutions to the fast diffusion equation
$u_t = \Delta u^m$ on $\R^N\times (0,T)$, in the range of exponents $m \in (0,
\frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in
between two Barenblatt solutions vanishing at time $T$, then the vanishing
behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an
example showing that for such a behavior the bound from above by a Barenblatt
solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions
$u$ with initial data $u_0 = B (1 + o(1))$, near $ |x| >> 1$, which live longer
than $B$ and change behaviour at $T$. The behavior of such solutions is
governed by $B(\cdot,t)$ up to $T$, while for $t >T$ the solutions become
integrable and exhibit a different vanishing profile. For the Yamabe flow ($m =
\frac{N-2}{N+2}$) the above means that these solutions $u$ develop a
singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$
they immediately smoothen up and exhibit the vanishing profile of a sphere.
  In the appendix we show how to remove the assumption on the bound on $u_0$
from below by a Barenblatt.
