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We investigate fine global properties of nonnegative, integrable solutions tothe Cauchy problem for the Fast Diffusion Equation with weights (WFDE)$u_t=|x|^\gamma\mathrm{div}\left(|x|^{-\beta}\nabla u^m\right)$ posed on$(0,+\infty)\times\mathbb{R}^d$, with $d\ge 3$, in the so-called good fastdiffusion range $m_c<m<1$, within the range of parameters $\gamma, \beta$,optimal for the validity of the so-called Caffarelli-Kohn-Nirenberginequalities. It is a natural question to ask in which sense such solutionsbehave like the Barenblatt $\mathfrak{B}$ (fundamental solution): for instance,asymptotic convergence, i.e. $\|u(t)-\mathfrak{B}(t)\|_{{\rmL}^p(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$, is well known for all $1\lep\le \infty$, while only few partial results tackle a finer analysis of thetail behaviour. We characterize the maximal set of data $\mathcal{X}\subset{\rmL}^1_+(\mathbb{R}^d)$ that produces solutions which are pointwise trappedbetween two Barenblatt (Global Harnack Principle), and uniformly converge inrelative error (UREC), i.e. ${\rmd}_\infty(u(t))=\|u(t)/\mathcal{B}(t)-1\|_{{\rmL}^\infty(\mathbb{R}^d)}\xrightarrow[]{t\to\infty}0$. Such characterization isin 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 clearanswers. Our characterization is also new for the classical, non-weighted, FDE.We are able to provide minimal rates of convergence to $\mathcal{B}$ indifferent norms. Such rates are almost optimal in the non weighted case, andbecome optimal for radial solutions. To complete the panorama, we show thatsolutions with data in ${\rm L}^1_+(\mathbb{R}^d)\setminus\mathcal{X}$,preserve the same "fat" spatial tail for all times, hence UREC fails.

Fine properties of solutions to the Cauchy problem for a Fast Diffusion Equation with Caffarelli-Kohn-Nirenberg weights