  Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positive
solutions to the fast diffusion equation $u_t=\Delta u^m$ in
$\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scaling
parameter $\beta$ inspired by \cite{DKS}, we study the second-order asymptotics
of the self-similar solutions associated with $\beta$ at spatial infinity. We
also investigate the asymptotic behavior of the solutions to the fast diffusion
equation near the vanishing time $T$, provided that the initial value of the
solution is close to the initial value of some self-similar solution and
satisfies some proper decay condition at infinity. Depending on the range of
the parameter $\beta$, we prove that the rescaled solution converges either to
a self-similar profile or to zero as $t\nearrow T$. The former implies
asymptotic stabilization towards a self-similar solution, and the latter is a
new vanishing phenomenon even for the case $n\ge3$ and $m=\frac{n-2}{n+2}\,$
which corresponds to the Yamabe flow on $\mathbb{R}^n$ with metric
$g=u^{\frac{4}{n+2}}dx^2$.
