We study the extinction behaviour of solutions to the fast diffusion equation ut = Δum on ℝN × (0, T), in the range of exponents . We show that if the initial value u0 is trapped in between two Barenblatt solutions vanishing at time T, then the vanishing behaviour of u at T is given by a Barenblatt solution. We also give an example showing that for such a behaviour the bound from above by a Barenblatt solution B (vanishing at T) is crucial: we construct a class of solutions u with initial value u0 = B(1 + o(1)), near |x| » 1, which live longer than B and change behaviour at T. The behaviour of such solutions is governed by B(·, t) up to T, while for t > T the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow the above means that these solutions u develop a singularity at time T, when the Barenblatt solution disappears, and at t > T they immediately smoothen up and exhibit the vanishing profile of a sphere.
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