Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) u(t) = vertical bar x vertical bar(gamma) div(vertical bar x vertical bar(-beta) del u(m) )posed on (0, + infinity) x R-d, with d >= 3, in the so-called good fast diffusion range m(c) < m < 1, within the range of parameters gamma, beta which is optimal for the validity of the so-called Caffarelli-Kohn-Nirenberg inequalities. It is natural to ask in which sense such solutions behave like the Barenblatt B (fundamental solution): for instance, asymptotic convergence, i.e vertical bar vertical bar u(t) - B(t)vertical bar vertical bar(Lp(Rd)) (t ->infinity)under right arrow 0, is well known for all 1 <= p <= infinity, while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X subset of L-+(1) (R-d)that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. d(infinity)(u(t) = vertical bar vertical bar u(t)/B(t) - 1 vertical bar vertical bar(L infinity(Rd)) (t ->infinity)under right arrow 0. Such a characterization is in terms of an integral condition on u(t = 0). To the best of our knowledge, analogous issues for the linear heat equation, m = 1, do not possess such clear answers, but only partial results. Our characterization is also new for the classical, nonweighted FDE. We are able to provide minimal rates of convergence to B in different norms. Such rates are almost optimal in the nonweighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in L-+(1) (R-d) / X, preserve the same "fat" spatial tail for all times, hence UREC fails and d(infinity) (u(t)) = infinity, even if vertical bar vertical bar u(t) - B(t)vertical bar vertical bar(L1(Rd)) (t ->infinity)under right arrow 0.