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Strongly anisotropic type II blow up at an isolated point

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Collot, Charles 
Merle, Frank 
Raphaël, Pierre 

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.

Description

Keywords

4901 Applied Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

Journal of the American Mathematical Society

Conference Name

Journal ISSN

0894-0347
1088-6834

Volume Title

Publisher

American Mathematical Society (AMS)

Rights

All rights reserved
Sponsorship
European Commission Horizon 2020 (H2020) ERC (SINGWAVES 646650)
ERC consolidator grant SingWaves