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On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation
About this Title
Charles Collot, Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, France, Pierre Raphaël, Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, France and Jeremie Szeftel, Laboratoire Jacques-Louis Lions, Université Paris 6, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 260, Number 1255
ISBNs: 978-1-4704-3626-1 (print); 978-1-4704-5334-3 (online)
DOI: https://doi.org/10.1090/memo/1255
Published electronically: July 23, 2019
MSC: Primary 35B32, 35B35, 35B44, 35J61, 35K58
Table of Contents
Chapters
- 1. Introduction
- 2. Construction of self-similar profiles
- 3. Spectral gap in weighted norms
- 4. Dynamical control of the flow
- A. Coercivity estimates
- B. Proof of (4.43)
- C. Proof of Lemma
- D. Proof of Lemma
Abstract
We consider the energy super critical semilinear heat equation \begin{equation*} \partial _tu=\Delta u+u^{p}, \ \ x\in \Bbb R^3, \ \ p>5. \end{equation*} We first revisit the construction of radially symmetric self similar solutions performed through an ode approach in Troy (1987), Budd and Qi (1989), and propose a bifurcation type argument suggested in Biernat and Bizon (2011) which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.- Haïm Brezis and Thierry Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304. MR 1403259, DOI 10.1007/BF02790212
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