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The damped focusing cubic wave equation on a bounded domain

Thomas Perrin

Abstract

For the focusing cubic wave equation on a compact Riemannian manifold of dimension $3$, the dichotomy between global existence and blow-up for solutions starting below the energy of the ground state is known since the work of Payne and Sattinger. In the case of a damped equation, we prove that the dichotomy between global existence and blow-up still holds. In particular, the damping does not prevent blow-up. Assuming that the damping satisfies the geometric control condition, we then prove that any global solution converges to a stationary solution along a time sequence, and that global solutions below the energy of the ground state can be stabilised, adapting the proof of a similar result in the defocusing case.

The damped focusing cubic wave equation on a bounded domain

Abstract

For the focusing cubic wave equation on a compact Riemannian manifold of dimension , the dichotomy between global existence and blow-up for solutions starting below the energy of the ground state is known since the work of Payne and Sattinger. In the case of a damped equation, we prove that the dichotomy between global existence and blow-up still holds. In particular, the damping does not prevent blow-up. Assuming that the damping satisfies the geometric control condition, we then prove that any global solution converges to a stationary solution along a time sequence, and that global solutions below the energy of the ground state can be stabilised, adapting the proof of a similar result in the defocusing case.
Paper Structure (24 sections, 23 theorems, 200 equations)

This paper contains 24 sections, 23 theorems, 200 equations.

Table of Contents

  1. Outline of the article.
  2. Acknowledgements.
  3. Preliminaries
  4. The Cauchy problem and some properties of the linear equation
  5. Ground state and some related properties
  6. Step 1: an explicit Sobolev embedding.
  7. Step 2: estimation of $\Vert u \Vert_{H_0^1}$.
  8. Step 3: end of the proof.
  9. The damped equation below the energy of the ground state
  10. Global solutions
  11. Blow-up solutions
  12. Case 1: bounded energy.
  13. Case 2: the energy tends to $- \infty$.
  14. Convergence towards a stationary solution
  15. Convergence of a bounded sequence along a time sequence
  16. ...and 9 more sections

Key Result

Theorem 1

The spaces $\mathscr{K}^+$ and $\mathscr{K}^-$ are stable under the flow of (KG_nL). A solution starting from $\mathscr{K}^+$ is defined on $\mathbb{R}$, and a solution starting from $\mathscr{K}^-$ blows up in finite positive and negative times.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Theorem 10: Strichartz estimates
  • ...and 29 more