| Abstract: |
| Let $n\ge 3$, $00$, I prove the existence and uniqueness (for $\beta\ge\frac{\rho_1}{n-2-nm}$) of radially symmetric singular solution $g_{\lambda}\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$ of the elliptic equation $\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\lim_{|x|\to 0}|x|^{\alpha/\beta}g_{\lambda}(x)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}$. When $\beta$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|\to\infty$. I also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence I prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=\Delta u^m$ in $\R^n\times (0,T)$ near the extinction time $T>0$. |
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