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 $\tilde{u}(x,t)=e^{\frac{2\beta }{1-m}t}u(e^{\beta t}x,t)$ converges uniformly on every compact subsets of ${\mathbb{R}}^n$ to $v_{{\lambda }_1,\beta }$ for some constant ${\lambda }_1>0$.

15 pages, introduction is rewritten, some typo corrected and some lemmas added