  We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any
$n\ge 3$, $0<m<\frac{n-2}{n+2}$ and $\gamma>0$, we will construct subsolutions
and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$
in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate
$(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the
existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}\Delta
u^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$,
which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$ when
$u_0$ satisfies appropriate decay condition.
