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Product Code:  MEMO/267/1301 
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Softcover ISBN:  9781470442996 
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Softcover ISBN:  9781470442996 
Product Code:  MEMO/267/1301 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470464011 
Product Code:  MEMO/267/1301.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470442996 
eBook ISBN:  9781470464011 
Product Code:  MEMO/267/1301.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 267; 2020; 129 ppMSC: Primary 35
The author shows 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 codimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter \(\lambda (t) = t^1\nu \) is sufficiently close to the selfsimilar rate, i. e. \(\nu >0\) is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ \partial _t^2 + \partial _r^2 + \frac 2r\partial _r +V(\lambda (t)r) \] for suitable monotone scaling parameters \(\lambda (t)\) and potentials \(V(r)\) with a resonance at zero.

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 reiteration

12. Convergence of the iterative scheme

13. Proof of the main technical theorem

14. Appendix


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The author shows 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 codimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter \(\lambda (t) = t^1\nu \) is sufficiently close to the selfsimilar rate, i. e. \(\nu >0\) is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ \partial _t^2 + \partial _r^2 + \frac 2r\partial _r +V(\lambda (t)r) \] for suitable monotone scaling parameters \(\lambda (t)\) and potentials \(V(r)\) with a resonance at zero.

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 reiteration

12. Convergence of the iterative scheme

13. Proof of the main technical theorem

14. Appendix