On the asymptotic behaviour of some solutions of the fast diffusion equation.

The so-called Barenblatt profiles are known to be explicit solution to the fast diffusion equation. Such solutions play, in the study of such equation, a role similar to the one played by the Gaussian solutions when dealing with the heat equation. In this talk we shall in fact show that, in suitable senses, certain classes of solutions to the fast diffusion equation converge to the Barenblatt profiles, giving explicit rates of convergence. Such rates are related to the best constant, explicitly determined, in a suitable Hardy-Poincaré inequality. A particular case, in which such constant vanishes, shows a polynomial decay instead of an exponential one. Such behaviour is proved using a geometric interpretation of the linearized evolution, the Li-Yau theory on the heat kernel on manifolds with nonnegative Ricci curvature, some weighted Nash inequalities and, finally, an appropriate use of the parabolic Harnack inequality.

This is a report of joint works with M. Bonforte, J.L. Vazquez and, in part, A. Blanchet e J. Dolbeault.