Type II blow up manifolds for the energy supercritical semilinear wave equation / Charles Collot.

Format:

Book

Author/Creator:

Collot, Charles, 1990- author.

Series:

Memoirs of the American Mathematical Society ; Volume 252, Number 1205.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 252, Number 1205

Language:

English

Subjects (All):

Wave equation.

Manifolds (Mathematics).

Physical Description:

1 online resource (176 pages).

Edition:

1st ed.

Other Title:

Type two blow up manifolds for the energy supercritical semilinear wave equation

Place of Publication:

Providence, RI : American Mathematical Society, [2018]

Summary:

Click here to view the abstract.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. The Semilinear Wave Equation

1.2. Blow up for (NLW)

1.3. Statement of the Result

1.4. Strategy of the Proof

Chapter 2. The Linearized Dynamics and the Construction of the Approximate Blow up Profile

2.1. The Stationary State and its Numerology

2.2. Factorization of ℒ

2.3. Inverting \bos{ } on Radially Symmetric Functions

2.4. Adapted Derivatives, Admissible and Homogeneous Functions

2.5. Slowly Modulated Blow up Profiles and Growing Tails

2.6. Study of the Dynamical System Driving the Evolution of the Parameters ( ᵢ)_{1≤ ≤ }

Chapter 3. The Trapped Regime

3.1. Setting up the Bootstrap

3.1.1. Projection onto the Approximate Solutions Manifold

3.1.2. Modulation

3.1.3. Adapted Norms

3.1.4. Estimates of the Bootstrap and Main Proposition

3.2. Evolution Equations for \bos{ } and \bos{ }

3.3. Modulation Equations

3.4. Improved Modulation Equation for _{ }

3.5. Lyapunov Monotonicity for the Low Sobolev Norm

3.6. Lyapunov Monotonicity for the High Sobolev Norm

3.7. Control from a Morawetz Type Quantity

Chapter 4. End of the Proof

4.1. End of the Proof of Proposition 3.2

4.2. Behavior of Sobolev Norms near Blow up Time

Chapter 5. Lipschitz Aspect and Codimension of the Set of Solutions Described by Proposition 3.2

5.1. Lipschitz Dependence of the Unstable Modes under Extra Assumptions

5.1.1. Adapted Time for Comparison, Notations and Identities

5.1.2. Modulation Equations for the Difference

5.1.3. Energy Identities for the Difference of Errors

5.1.4. Study of the Coupled Dynamical System, End of the Proof of Proposition \fref{variete:prop:parametres lipschitz}

5.2. Removal of Extra Assumptions, End of the Proof of Theorem 5.1

5.2.1. Lower Order Decomposition.

Appendix A. Properties of the Stationary State

Appendix B. Equivalence of Norms

Appendix C. Hardy Inequalities

Appendix D. Coercivity of the Adapted Norms

Appendix E. Specific Bounds for the Analysis

Acknowledgment

Bibliography

Index

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-4379-1