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Scattering property for a system of Klein-Gordon equations with energy below ground state

Yan Cui, Bo Xia

Abstract

In the previous work, we classified the solutions to a family of systems of Klein-Gordon equations with non-negative energy below the ground state into two parts: one blows up in finite time while the other extends to a global solution. In the present work, we strengthen this result, showing that these global solutions are indeed scattering in the energy space. Here we adapted Kenig-Merle's concentration-compactness approach to the system.

Scattering property for a system of Klein-Gordon equations with energy below ground state

Abstract

In the previous work, we classified the solutions to a family of systems of Klein-Gordon equations with non-negative energy below the ground state into two parts: one blows up in finite time while the other extends to a global solution. In the present work, we strengthen this result, showing that these global solutions are indeed scattering in the energy space. Here we adapted Kenig-Merle's concentration-compactness approach to the system.
Paper Structure (14 sections, 19 theorems, 211 equations)

This paper contains 14 sections, 19 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces and Strichartz estimates
  4. Basic Cauchy theory
  5. Lorentz symmetries
  6. Variational results
  7. Linear profile decomposition
  8. Perturbation lemma
  9. Proof of main result
  10. Extraction of a critical element
  11. Compactness about the critical element
  12. The $0$-momentum property of the critical element
  13. Improvement of the growth of $x_0(t)$
  14. Arriving at the contradiction

Key Result

Theorem 1.1

With the notations as above and $\mathcal{H}:=H^1\times L^2$, for each $\beta\in[0,\infty)$, both regions defined by and are invariant under the flow of eq:skg:int, as long as the flow is defined. What's more, arguing in the spirit of Payne-Sattinger yields the following dichotomy:

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1: nakanishischlag2011
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 35 more