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Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

Ruipeng Shen

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. It has been verified that any global or type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a small error, as time tends to infinity or the blow-up time. This is usually called soliton resolution. However, all known examples of soliton resolution in the 3D radial case come with no more than one soliton. In this work we prove the nonexistence of any global or type II blow up solution with two or more solitons, thus give a complete classification of asymptotic behaviours of radial solutions.

Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. It has been verified that any global or type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a small error, as time tends to infinity or the blow-up time. This is usually called soliton resolution. However, all known examples of soliton resolution in the 3D radial case come with no more than one soliton. In this work we prove the nonexistence of any global or type II blow up solution with two or more solitons, thus give a complete classification of asymptotic behaviours of radial solutions.
Paper Structure (39 sections, 21 theorems, 280 equations)

This paper contains 39 sections, 21 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Finite time blow-up
  4. Soliton resolution
  5. Number of bubbles
  6. Main topic and result
  7. Type II blow-up solutions
  8. Global solutions
  9. Main result
  10. General idea
  11. Radiation fields
  12. Radiation concentration
  13. Bubble interaction with no dispersion
  14. Major tools
  15. Structure of this work
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $u$ be a radial solution of (CP1) and $T_+ = T_+(u)$ be the right endpoint of this maximal interval of existence. Then one the following holds:

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 2.1: Radiation field
  • Lemma 2.2: Radiation fields of inhomogeneous equation
  • Remark 2.3
  • ...and 39 more