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Soliton resolution for the energy critical damped wave equations in the radial case

Jingyuan Gu, Lifeng Zhao

Abstract

We consider energy-critical damped wave equation \begin{equation*} \partial_{tt}u-Δu+α\partial_t u=\left|u\right|^{\frac{4}{D-2}}u \end{equation*} with radial initial data in dimensions $D\geq 4$. The equation has a nontrivial radial stationary solution $W$, called the ground state, which is unique up to sign and scale. We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term. In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect.

Soliton resolution for the energy critical damped wave equations in the radial case

Abstract

We consider energy-critical damped wave equation \begin{equation*} \partial_{tt}u-Δu+α\partial_t u=\left|u\right|^{\frac{4}{D-2}}u \end{equation*} with radial initial data in dimensions . The equation has a nontrivial radial stationary solution , called the ground state, which is unique up to sign and scale. We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term. In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect.
Paper Structure (26 sections, 49 theorems, 492 equations, 1 table)

This paper contains 26 sections, 49 theorems, 492 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Statement of the main result
  4. The outline of the proof
  5. Preliminaries
  6. Strichartz estimates and decay estimates
  7. Virial identities
  8. Multi-bubble configurations
  9. Linear profile decomposition
  10. Inversed Strichartz estimate
  11. Proof of the linear profile decomposition
  12. Sequential soliton resolution
  13. Sequential compactness lemma
  14. Extraction of the radiation
  15. The sequential decomposition
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $D \geq 4$ and let $\boldsymbol{u}(t)$ be a finite energy solution to (DNLW) with initial data $\boldsymbol{u}(0)=\boldsymbol{u}_0\in \mathcal{E},$ defined on its maximal forward interval of existence $[0,T_+)$. Suppose that Then, (Global solution) if $T_+=\infty$, there exists a time $T_0>0$, an integer $N\geq 0$, continuous functions $\lambda_1(t),...,\lambda_N(t)\in C^0([T_0,T_+)),$ signs

Theorems & Definitions (79)

  • Theorem 1.1: Soliton Resolution
  • Remark 1.1
  • Proposition 1.1: Properties of the radiation, finite time case
  • Proposition 1.2: Properties of the radiation, global case
  • Definition 1.1: Multi-bubble configuration
  • Theorem 1.2: Sequential soliton resolution
  • Definition 2.1: solution
  • Lemma 2.1: $L^p-L^q$ estimates
  • Lemma 2.2: Homogeneous Strichartz estimates Inui
  • Lemma 2.3: Inhomogeneous Strichartz estimatesInui
  • ...and 69 more