  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 $t\to\infty$ for $m\ge m_c=(d-2)/d$, 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.
