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=|x|^(|x|-β∇ u^m) posed on (0,+∞)^d , with d≥ 3 , in the so-called good fast diffusion range m_c<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 natural to ask in which sense such solutions behave like the Barenblatt B (fundamental solution): for instance, asymptotic convergence, i.e. \|u(t)-B(t)\|L^p(R^d){t→∞}0 , is well known for all 1≤ p≤ ∞ , while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X^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_∞(u(t))=\|u(t)/B(t)-1\|L^∞(R^d){t→∞}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) , preserve the same “fat” spatial tail for all times, hence UREC fails and d_∞(u(t)) = ∞ , even if \|u(t)-B(t)\|L^1(R^d){t→∞} 0 .