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

For \begin{document}$ n\ge 3 $\end{document} , \begin{document}$ 0<m<\frac{n-2}{n} $\end{document} , \begin{document}$ \beta<0 $\end{document} and \begin{document}$ \alpha = \frac{2\beta}{1-m} $\end{document} , we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in \begin{document}$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $\end{document} of the form \begin{document}$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $\end{document} where \begin{document}$ f_{\lambda} $\end{document} is a radially symmetric function satisfying\begin{document}$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $\end{document}with \begin{document}$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $\end{document} and \begin{document}$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $\end{document} , for some constant \begin{document}$ \lambda>0 $\end{document} . As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $\end{document} with initial value \begin{document}$ u_0 $\end{document} satisfying \begin{document}$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $\end{document} , \begin{document}$ \forall x\in\mathbb{R}^n\setminus\{0\} $\end{document} , such that the solution \begin{document}$ u $\end{document} satisfies \begin{document}$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $\end{document} , \begin{document}$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $\end{document} , for some constants \begin{document}$ \lambda_1>\lambda_2>0 $\end{document} . We also prove the asymptotic large time behaviour of such singular solution \begin{document}$ u $\end{document} when \begin{document}$ n = 3,4 $\end{document} and \begin{document}$ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $\end{document} holds. Asymptotic large time behaviour of such singular solution \begin{document}$ u $\end{document} is also obtained when \begin{document}$ 3\le n<8 $\end{document} , \begin{document}$ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $\end{document} , and \begin{document}$ u(x,t) $\end{document} is radially symmetric in \begin{document}$ x\in\mathbb{R}^n\setminus\{0\} $\end{document} for any \begin{document}$ t>0 $\end{document} under appropriate conditions on the initial value \begin{document}$ u_0 $\end{document} .