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Journal of the American Mathematical Society

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Strongly anisotropic type II blow up at an isolated point


Authors: Charles Collot, Frank Merle and Pierre Raphaël
Journal: J. Amer. Math. Soc. 33 (2020), 527-607
MSC (2010): Primary 35K58
DOI: https://doi.org/10.1090/jams/941
Published electronically: February 20, 2020
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Abstract: We consider the energy supercritical $ d+1$-dimensional semi-linear heat equation

$\displaystyle \partial _tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$    

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
Email: ccollot@unice.fr

Frank Merle
Affiliation: Laboratoire AGM, Université de Cergy Pontoise, France –and– Institute des Hautes Études Scientifiques
Email: merle@math.u-cergy.fr

Pierre Raphaël
Affiliation: Laboratoire J.A. Dieudonné, Université de la Côte d’Azur, France
Email: praphael@unice.fr

DOI: https://doi.org/10.1090/jams/941
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.
Article copyright: © Copyright 2020 American Mathematical Society