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On stability of type II blow up for the critical nonlinear wave equation on $\mathbb {R}^{3+1}$
About this Title
Joachim Krieger
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 267, Number 1301
ISBNs: 978-1-4704-4299-6 (print); 978-1-4704-6401-1 (online)
DOI: https://doi.org/10.1090/memo/1301
Published electronically: January 4, 2021
Keywords: Critical wave equation,
blowup
Table of Contents
Chapters
- 1. Introduction
- 2. Recall of the construction of $u_{\nu }$ and the distorted Fourier transform
- 3. Growth properties of the forward linear parametrix
- 4. Setting up the perturbative problem
- 5. Nonlinear estimates
- 6. Outline of the iterative scheme
- 7. Control of the first iterate; contribution of the linear terms $\mathcal {R}(\tau , \underline {x}^{(0)})$
- 8. Control of the first iterate; contribution of the nonlinear terms
- 9. Iterative step; a priori control of higher iterates
- 10. Preparations for the proof of convergence; refined estimates
- 11. Improvements upon re-iteration
- 12. Convergence of the iterative scheme
- 13. Proof of the main technical theorem
- 14. Appendix
Abstract
We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^{5} \] on $\mathbb {R}^{3+1}$ constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $\lambda (t) = t^{-1-\nu }$ is sufficiently close to the self-similar rate, i. e. $\nu >0$ is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -∂_{t}^{2} + ∂_{r}^{2} + \frac2r∂_{r} +V(𝜆(t)r) \] for suitable monotone scaling parameters $\lambda (t)$ and potentials $V(r)$ with a resonance at zero.- Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. MR 1705001
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