We investigate fine global properties of nonnegative, integrable solutions to the 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 fast diffusion range $m_c<m<1$, within the range of parameters $\gamma, \beta$, 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 $\mathfrak{ B}$ (fundamental solution): for instance, asymptotic convergence, i.e. $\|u(t)-\mathfrak{ B}(t)\|_{ { \rm L}^p(\mathbb{ R}^d)}\xrightarrow[]{ t\to\infty}0$, is well known for all $1\le p\le \infty$, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $\mathcal{ X}\subset{ \rm L}^1_+(\mathbb{ R}^d)$ that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (UREC), i.e. ${ \rm d}_\infty(u(t))=\|u(t)/\mathcal{ B}(t)-1\|_{ { \rm L}^\infty(\mathbb{ R}^d)}\xrightarrow[]{ t\to\infty}0$. 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. Our characterization is also new for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to $\mathcal{ B}$ 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 ${ \rm L}^1_+(\mathbb{ R}^d)\setminus\mathcal{ X}$, preserve the same "fat" spatial tail for all times, hence UREC fails.
Categories: math.AP