On the extinction profile of solutions to fast diffusion

Panagiota Daskalopoulos, Natasa Sesum

Research output: Contribution to journalArticle

31 Scopus citations

Abstract

We study the extinction behaviour of solutions to the fast diffusion equation ut = Δum on ℝN × (0, T), in the range of exponents m ε (0, N-2/N), N ≥ 2. 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 (m = N - 2/N + 2)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.

Original languageEnglish (US)
Pages (from-to)95-119
Number of pages25
JournalJournal fur die Reine und Angewandte Mathematik
Issue number622
DOIs
StatePublished - Sep 1 2008
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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