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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Strongly anisotropic type II blow up at an isolated point
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by Charles Collot, Frank Merle and Pierre Raphaël HTML | PDF
J. Amer. Math. Soc. 33 (2020), 527-607 Request permission

Abstract:

We consider the energy supercritical $d+1$-dimensional semi-linear heat equation \begin{equation*} \partial _tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14. \end{equation*} A fundamental open problem on this canonical nonlinear model is to understand the possible blow-up profiles appearing after renormalisation of a singularity. We exhibit in this paper a new scenario corresponding to the first example of a strongly anisotropic blow-up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the reconnection problem in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilisation of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow-up speeds.
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Additional Information
  • Charles Collot
  • Affiliation: Laboratoire J.A. Dieudonné, Université de la Côte d’Azur, France
  • MR Author ID: 1192817
  • Email: ccollot@unice.fr
  • Frank Merle
  • Affiliation: Laboratoire AGM, Université de Cergy Pontoise, France –and– Institute des Hautes Études Scientifiques
  • MR Author ID: 123710
  • Email: merle@math.u-cergy.fr
  • Pierre Raphaël
  • Affiliation: Laboratoire J.A. Dieudonné, Université de la Côte d’Azur, France
  • Email: praphael@unice.fr
  • Received by editor(s): May 17, 2017
  • Received by editor(s) in revised form: May 2, 2019, and September 9, 2019
  • Published electronically: February 20, 2020
  • Additional Notes: The first and third authors were supported by the ERC-2014-CoG 646650 SingWave.
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 527-607
  • MSC (2010): Primary 35K58
  • DOI: https://doi.org/10.1090/jams/941
  • MathSciNet review: 4073868