Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, ${\rho }_1>0$, $\beta \ge \frac{m{\rho }_1}{n-2-nm}$ and $\alpha =\frac{2\beta +{\rho }_1}{1-m}$. For any $\lambda >0$, we will prove the existence and uniqueness (for $\beta \ge \frac{{\rho }_1}{n-2-nm}$) of radially symmetric singular solution $g_{\lambda }\in C^{\mathrm{\infty }}(R^n\setminus \{0\})$ of the elliptic equation $\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$,$v>0$,in$R^n\setminus{0}$,satisfying$\displaystyle\lim_{|x|\to 0}|x|^{\alpha/\beta}g_{\lambda}(x)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}$\mathrm{.}When$\beta$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=\mathrm{\Delta }{u}^m$ in $R^n\times (0,T)$ near the extinction time $T>0$.