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 $g=u^{ \frac{ 4}{ n+2}}dx^2$.
Categories: math.AP