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Nonexistence of pure bubble type II blow-up solutions to energy critical wave equation in the 3D radial case

Ruipeng Shen

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. According to the soliton resolution conjecture, which has been verified in the radial case, any 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 the blow-up time. We prove that there does not exist any pure bubble type II blow up solutions. In other words, the free wave part is never zero in the soliton resolution of type II blow-up solutions, regardless of the bubble number.

Nonexistence of pure bubble type II blow-up solutions to energy critical wave equation in the 3D radial case

Abstract

In this work we consider the focusing, energy-critical wave equation in 3D radial case. According to the soliton resolution conjecture, which has been verified in the radial case, any 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 the blow-up time. We prove that there does not exist any pure bubble type II blow up solutions. In other words, the free wave part is never zero in the soliton resolution of type II blow-up solutions, regardless of the bubble number.
Paper Structure (31 sections, 13 theorems, 205 equations, 1 figure)

This paper contains 31 sections, 13 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Ground states
  4. Finite time blow-up
  5. Type II blow-up solution
  6. Existence of type II blow-up solutions
  7. Main topic and result
  8. General idea
  9. Extraction of radiation
  10. Soliton resolution
  11. Extraction of good times
  12. Inserting the virial identity
  13. Structure of this work
  14. Preliminary results
  15. Notations
  16. ...and 16 more sections

Key Result

Theorem 1.1

There does not exist any radial pure bubble type II blow-up solution $u$ to (CP1). In other words, the radiation part $u_L$ in the soliton resolution above can never be zero for any radial finite-time type II blow-up solution.

Figures (1)

  • Figure 1: Domain extension of $u$

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1: Radiation field
  • Lemma 2.2: Radiation fields of inhomogeneous equation
  • Remark 2.3
  • Proposition 2.4
  • ...and 21 more