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

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $ρ_1>0$, $β\ge\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. For any $λ>0$, we will prove the existence and uniqueness (for $β\ge\frac{ρ_1}{n-2-nm}$) of radially symmetric singular solution $g_λ\in C^{\infty}(R^n\setminus\{0\})$ of the elliptic equation $Δv^m+αv+βx\cdot\nabla v=0$, $v>0$, in $R^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}g_λ(x)=λ^{-\frac{ρ_1}{(1-m)β}}$. When $β$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|\to\infty$. 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 $u_t=Δu^m$ in $R^n\times (0,T)$ near the extinction time $T>0$.