Fine asymptotics for fast diffusion equations

We investigate the large-time asymptotics of fast diffusion equations, u(1) = Deltau(m) where 0 < m < 1. We calculate convergence to Barenblatt profiles with algebraic rates in the exponent interval (d-2)/d < m < (d-1)/d in dimensions d greater than or equal to 2. We cover in this way the gap still existing in the literature concerning the rates of approach to a Barenblatt profile, which have been recently obtained for all exponents m greater than or equal to (d-1)/d. Our main result improves the standard convergence parallel tou-Uparallel to(1) --> 0 into a rate of decay of the form parallel tou-Uparallel to(1) = O(t(-1/2)), t --> infinity, valid for any (d-2)/d < m < 1 and any solution with nonnegative and integrable data satisfying a condition of finite relative entropy. In the formula it is the actual solution, U the asymptotic model, and the L-1 norm is taken in the space variable. Let us recall that the Barenblatt profiles do not exist for m less than or equal to (d-2)/d and then the solutions have a quite different large-time evolution. We are also concerned with the presence of certain critical exponents in the asymptotic behavior; we explain the role of the value (d-I)/d, by looking at the linearized equation. Finally, we obtain a better rate of convergence of the form (u-U)/U = O(t(-1)) for radially symmetric solutions with strongly decaying data as \x\ --> infinity. This rate is optimal.