Asymptotics of the fast diffusion equation via entropy estimates

We consider non-negative solutions of the fast diffusion equation $u_t=\Delta u^m$with$m \in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to $t\to \mathrm{\infty }$ for $m\ge m_c=(d-2)\mathrm{/}d$, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if $m\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results.