Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space

For $nge 3$, $0<m<frac{n-2}{n}$, $beta<0$ and $alpha=frac{2beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in $(mathbb{R}^nsetminus{0})times mathbb{R}$ of the form $U_{lambda}(x,t)=e^{-alpha t}f_{lambda}(e^{-beta t}x), xin mathbb{R}^nsetminus{0}, tinmathbb{R},$ where $f_{lambda}$ is a radially symmetric function satisfying $$frac{n-1}{m}Delta f^m+alpha f+beta xcdot abla f=0 text{ in }mathbb{R}^nsetminus{0},$$ with $underset{substack{rto 0}}{lim}frac{r^2f(r)^{1-m}}{log r^{-1}}=frac{2(n-1)(n-2-nm)}{|beta|(1-m)}$ and $underset{substack{rtoinfty}}{lim}r^{frac{n-2}{m}}f(r)=lambda^{frac{2}{1-m}-frac{n-2}{m}}$, for some constant $lambda>0$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=frac{n-1}{m}Delta u^m$ in $(mathbb{R}^nsetminus{0})times (0,infty)$ with initial value $u_0$ satisfying $f_{lambda_1}(x)le u_0(x)le f_{lambda_2}(x)$, $forall xinmathbb{R}^nsetminus{0}$, which satisfies $U_{lambda_1}(x,t)le u(x,t)le U_{lambda_2}(x,t)$, $forall xin mathbb{R}^nsetminus{0}, tge 0$, for some constants $lambda_1>lambda_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $ttoinfty$ when $n=3,4$ and $frac{n-2}{n+2}le m<frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $ttoinfty$ is also obtained when $3le n<8$, $1-sqrt{2/n}le m<minleft(frac{2(n-2)}{3n},frac{n-2}{n+2}right)$, and $u(x,t)$ is radially symmetric in $xinmathbb{R}^nsetminus{0}$ for any $t>0$ under appropriate conditions on the initial value $u_0$.