Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non–linear Fokker–Planck equation, we find a continuum of algebraic rates of convergence to a self–similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631–680, 2010) and provides answers to some open problems.