Abstract (EN)

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) ut = |x|γdiv|x|−β∇um) posed on (0, +∞)×Rd, with d ≥ 3, in the so-called good fast diffusion range mc < m < 1, within the range of parameters γ, β which is optimal for the validity of the so-called Caffarelli-Kohn-Nirenberg inequalities. It is a natural question to ask in which sense such solutions behave like the Barenblatt B (fundamental solution): for instance, asymptotic convergence, i.e. ku(t) − B(t)kLp(Rd) t→∞ −−−→ 0, is well known for all 1 ≤ p ≤ ∞, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X ⊂ L1+(Rd) that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (UREC), i.e. d((t))=ku(t)/B(t) − 1kL∞(Rd)t→∞ −−−→ 0. Such 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, only partial results. Our characterization is also new for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to B in different norms. Such rates are almost optimal in the non weighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in L1+(Rd) \ X, preserve the same “fat” spatial tail for all times, hence UREC fails and d∞(u(t))=∞, even if ku(t) − B(t)kL1(Rd)t→∞ −−−→ 0