Type II Blow up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on {R}^{3+1}

Format:

Book

Author/Creator:

Burzio, Stefano.

Contributor:

Krieger, Joachim.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.278

Language:

English

Subjects (All):

Nonlinear wave equations.

Blowing up (Algebraic geometry).

Perturbation (Mathematics).

Asymptotic expansions.

Iterative methods (Mathematics).

Fourier transformations.

Physical Description:

1 online resource (88 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation on constructed in Krieger, Schlag, and Tartaru ("Slow blow-up solutions for the critical focusing semilinear wave equation", 2009) and Krieger and Schlag ("Full range of blow up exponents for the quintic wave equation in three dimensions", 2014) are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter is sufficiently close to the self-similar rate, i. e., is sufficiently small. This result is qualitatively optimal in light of the result of Krieger, Nakamishi, and Schlag ("Center-stable manifold of the ground state in the energy space for the critical wave equation", 2015). The paper builds on the analysis of Krieger and Wong ("On type I blow-up formation for the critical NLW", 2014)"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. The type II blow up solutions of [33], [32]

1.2. The effect of symmetries on the solutions of Theorem 1.1

1.3. Conditional stability of type II solutions

1.4. Spectral theory associated with the linearisation ℒ

1.5. Description of the data perturbation in terms of the distorted Fourier transform

1.6. Outline of the main result from [26]

1.7. Figures

Chapter 2. The main theorem and outline of the proof

2.1. The main theorem

2.2. Outline of the proof

Chapter 3. Construction of a two parameter family of approximate blow up solutions

3.1. Step 0: the bulk term

3.2. Step 1: choice of the first correction ₁

3.3. Step 2: the ₁ error

3.4. Step 3: choice of second correction ₂

3.5. Step 4: the ₂ error

3.6. Step 5: inductive step

3.7. Step 6: choice of _{ ℎ, }, =1,2

Chapter 4. Modulation theory

determination of the parameters _{1,2}.

4.1. Re-scalings and the distorted Fourier transform

4.2. The effect of scaling the bulk part

Chapter 5. Iterative construction of blow up solution almost matching the perturbed initial data

5.1. Formulation of the perturbation problem on Fourier side

5.2. The proof of Theorem 5.1

5.3. Translation to original coordinate system

Chapter 6. Proof of Theorem 2.1

Chapter 7. Outlook

Bibliography

Index

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Burzio, Stefano Type II Blow up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on {R}^{3+1}

ISBN:

9781470471699

1470471698

OCLC:

1336954728