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Center stable manifolds for the radial semi-linear wave equation outside a ball

Thomas Duyckaerts, Jianwei Urban Yang

Abstract

We consider the nonlinear wave equation, with a large exponent, power-like non-linearity, outside a ball of the Euclidean 3-dimensional space. In a previous article, we have proved that any global solution converges, up to a radiation term, to a stationary solution of the equation. In this work, we construct the center-stable manifold associated to each of the stationary solution, giving a complete description of the dynamics of global solutions. We also study the behavior of solutions close to each of the center-stable manifold.

Center stable manifolds for the radial semi-linear wave equation outside a ball

Abstract

We consider the nonlinear wave equation, with a large exponent, power-like non-linearity, outside a ball of the Euclidean 3-dimensional space. In a previous article, we have proved that any global solution converges, up to a radiation term, to a stationary solution of the equation. In this work, we construct the center-stable manifold associated to each of the stationary solution, giving a complete description of the dynamics of global solutions. We also study the behavior of solutions close to each of the center-stable manifold.
Paper Structure (31 sections, 34 theorems, 318 equations)

This paper contains 31 sections, 34 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Related works
  4. Outline of the article
  5. Linearized operators around stationary solutions
  6. Existence of stationary solutions
  7. Essential spectrum and the null space of $L_j$
  8. Existence of at least $j+1$ negative eigenvalues for $L_j$
  9. Existence of at most $j+1$ negative eigenvalues for $L_j$
  10. Reduction to counting zeros of $\Lambda Q_j$
  11. An auxiliary result via variational argument
  12. $\Lambda Q_j$ has $j+1$ zeros on $(1,+\infty)$
  13. Dispersive estimates
  14. Construction of the center-stable manifold
  15. The canonical decomposition of $\mathcal{H}$ and the projected linearized equation
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $(u_0,u_1)\in \mathcal{H}$ and $u$ be the corresponding solution of eq:NLW. Assume that $T_+(u)=+\infty$. Then there exists $Q\in \mathcal{Q}$ and a solution $u_L$ of the linear wave equation on $\Omega$: with initial data in $\mathcal{H}$, such that Furthermore,

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Global center stable manifold
  • Proposition 1.4: Tangent spaces
  • Proposition 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8: Blow-up manifold
  • Theorem 1.9: Stable blow-up
  • Theorem 1.10
  • ...and 63 more