On the extinction profile of solutions to fast diffusion

Panagiota Daskalopoulos 1 , 2  and Natasa Sesum 1 , 2
  • 1 Department of Mathematics, Columbia University, New York, USA. e-mail: natasas@math.columbia.edu
  • 2 Department of Mathematics, Columbia University, New York, USA. e-mail: pdaskalo@math.columbia.edu

Abstract

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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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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