Asymptotic large time behavior of singular solutions of the fast diffusion equation

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=\Delta u^m$ in $({\mathbb R}^n\setminus\{0\})\times(0,\infty)$ in the subcritical case $0<m<\frac{n-2}{n}$, $n\ge3$. Firstly, we prove the existence of singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-\alpha} f_i(t^{-\beta}x)$, $i=1,2$, with initial value $u_0$ satisfying $A_1|x|^{-\gamma}\le u_0\le A_2|x|^{-\gamma}$ for some constants $A_2>A_1>0$ and $\frac{2}{1-m}<\gamma<\frac{n-2}{m}$, where $\beta:=\frac{1}{2-\gamma(1-m)}$, $\alpha:=\frac{2\beta-1}{1-m},$ and the self-similar profile $f_i$ satisfies the elliptic equation $$ \Delta f^m+\alpha f+\beta x\cdot \nabla f=0\quad \mbox{in ${\mathbb R}^n\setminus\{0\}$} $$ with $\lim_{|x|\to0}|x|^{\frac{ \alpha}{ \beta}}f_i(x)=A_i$ and $\lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m}<\gamma<n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function $$ \tilde u(y,\tau):= t^{\,\alpha} u(t^{\,\beta} y,t),\quad{ \tau:=\log t}, $$ converges to some self-similar profile $f$ as $\tau\to\infty$.