Global behaviour of solutions of the fast diffusion equation

We will extend a recent result of B.~Choi and P.~Daskalopoulos (\cite{CD}). For any $n\ge 3$, $0<m<\frac{n-2}{n}$, $m\mathrm{\ne }\frac{n-2}{n+2}$, $\beta >0$ and $\lambda >0$, we prove the higher order expansion of the radially symmetric solution $v_{\lambda ,\beta }(r)$ of $\frac{n-1}{m}\Delta v^m+\frac{2\beta}{1-m} v+\beta x\cdot\nabla v=0$in$$\mathbb{R}$^n$,$v(0)=\lambda$,as$r\to\infty$. As a consequence for any $n\ge 3$ and $0<m<\frac{n-2}{n}$ if $u$ is the solution of the equation $u_t=\frac{n-1}{m}\mathrm{\Delta }{u}^m$ in $$\mathbb{R}$^n\times (0,\infty)$withinitialvalue$0\le u_0\in L^{\infty}($\mathbb{R}$^n)$satisfying$u_0(x)^{1-m}=\frac{2(n-1)(n-2-nm)}{(1-m)\beta |x|^2}\left(\log |x|-\frac{n-2-(n+2)m}{2(n-2-nm)}\log (\log |x|)+K_1+o(1))\right)$as$|x|\to\infty$forsomeconstants$\beta>0$and$K_1\in$\mathbb{R}$$,thenas$t\to\infty$therescaledfunction$\widetilde{u}(x,t)=e^{\frac{2\beta}{1-m}t}u(e^{\beta t}x,t)$ converges uniformly on every compact subsets of ${\mathbb{R}}^n$ to $v_{{\lambda }_1,\beta }$ for some constant ${\lambda }_1>0$.