Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation ut = Δum with m ∈ (0, 1) in the Euclidean space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}^d}$$\end{document}, d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ mc = (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 ≧ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≦ mc where previous approaches fail. In the range mc < m < 1, we improve on known results.