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

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\alpha =\frac{2\beta -1}{1-m}$ and $\frac{2}{1-m}<\frac{\alpha }{\beta }<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward self-similar solution of the form $V(x,t)=t^{-\alpha }f(t^{-\beta }x)$ $\mathrm{\forall }x\in {\mathbb{R}}^n\setminus \{0\}$, $t>0$, for the fast diffusion equation $u_t=\mathrm{\Delta }(u^m\mathrm{/}m)$ in $({\mathbb{R}}^n\setminus \{0\})\times (0,\mathrm{\infty })$, where $f$ satisfies \begin{equation*} Δ(f^m/m) + αf + βx \cdot \nabla f =0 \quad $\text{in}$ ; $\mathbb{R}$^n\setminus{0} \end{equation*} with ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to 0}\mathrm{\mid }x{\mathrm{\mid }}^{\frac{\alpha }{\beta }}f(x)=A$ and ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}f(x)=D_A$ for some constants $A>0$, $D_A>0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t=\mathrm{\Delta }(u^m\mathrm{/}m)$ in $({\mathbb{R}}^n\setminus \{0\})\times (0,\mathrm{\infty })$, $u(x,0)=u_0(x)$ in ${\mathbb{R}}^n\setminus \{0\}$, satisfying the condition $A_1\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }\le u_0(x)\le A_2\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }$ in ${\mathbb{R}}^n\setminus \{0\}$, for some constants $A_2>A_1>0$ and $n\le \gamma <\frac{n-2}{m}$.