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 $n\ge 3$, $0<m<\frac{n-2}{n}$, $m\ne\frac{n-2}{n+2}$, $\beta>0$ 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 $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (0,\infty)$ with initial value $0\le u_0\in L^{\infty}(\mathbb{R}^n)$ 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 $|x|\to\infty$ for some constants $\beta>0$ 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 $\lambda_1>0$.