Global behaviour of solutions of the fast diffusion equation

Abstract

We will extend a recent result of B.~Choi and P.~Daskalopoulos (\cite{CD}). For any $n\ge 3$, $0 0$ 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 $0 0$ and $K_1\in\mathbb{R}$, then as $t\to\infty$ the rescaled function $\widetilde{u}(x,t)=e^{\frac{2\beta}{1-m}t}u(e^{\beta t}x,t)$ converges uniformly on every compact subsets of $\mathbb{R}^n$ to $v_{\lambda_1,\beta}$ for some constant $\lambda_1>0$.