Strongly anisotropic type II blow up at an isolated point
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Authors
Collot, Charles
Merle, Frank
Raphaël, Pierre
Journal Title
Journal of the American Mathematical Society
ISSN
0894-0347
Publisher
American Mathematical Society (AMS)
Pages
1-1
Language
en
Type
Article
This Version
AM
Metadata
Show full item recordCitation
Collot, C., Merle, F., & Raphaël, P. (2019). Strongly anisotropic type II blow up at an isolated point. Journal of the American Mathematical Society, 1-1. https://doi.org/10.1090/jams/941
Abstract
We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\pa_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A fundamental open problem on this canonical nonlinear model is to understand the possible blow up profiles appearing after renormalization of a singularity. We exhibit in this paper a new scenario corresponding to the first example of 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 {\it 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 stabilization of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow up speeds.
Sponsorship
ERC consolidator grant SingWaves
Funder references
European Commission Horizon 2020 (H2020) ERC (SINGWAVES 646650)
Identifiers
External DOI: https://doi.org/10.1090/jams/941
This record's URL: https://www.repository.cam.ac.uk/handle/1810/301007
Rights
All rights reserved