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Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions

Alex Tullini

Abstract

We study the conformal wave equation $\square_g ψ+ \frac{2}{l^2} ψ= 0$ on 4-dimensional Schwarzschild--Anti de Sitter spacetimes under dissipative boundary conditions. We prove boundedness and decay of the non-degenerate energy of $ψ$ at an arbitrary polynomial rate of $(1+v)^{-n}$ provided that we control the (up to) $n$-times $T$-commuted energy. This contrasts with the inverse logarithmic decay obtained under Dirichlet boundary conditions and is in line with the result obtained in the pure Anti-de Sitter case under dissipative boundary conditions. In particular, the decay is not affected by the additional trapping at the photon sphere.

Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions

Abstract

We study the conformal wave equation on 4-dimensional Schwarzschild--Anti de Sitter spacetimes under dissipative boundary conditions. We prove boundedness and decay of the non-degenerate energy of at an arbitrary polynomial rate of provided that we control the (up to) -times -commuted energy. This contrasts with the inverse logarithmic decay obtained under Dirichlet boundary conditions and is in line with the result obtained in the pure Anti-de Sitter case under dissipative boundary conditions. In particular, the decay is not affected by the additional trapping at the photon sphere.

Paper Structure

This paper contains 24 sections, 15 theorems, 98 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The conformal wave equation in Anti-de Sitter space
  3. The exterior of Schwarzschild--Anti-de Sitter
  4. The conformal wave equation in Schwarzschild-AdS
  5. Main results
  6. Sketch of the proofs
  7. Further discussion
  8. Comparison with Lambda
  9. Outline of the paper
  10. Energy boundedness
  11. Energy boundedness for T energy
  12. Redshift Estimate
  13. Accessory notation & lemmata
  14. Proof of Proposition \ref{['prop:redshift']}
  15. Non-degenerate energy boundedness
  16. ...and 9 more sections

Key Result

Theorem 1.1

Fix $\overline{v}>v_0$. Given smooth functions $\psi_0,\psi_1:\Sigma_{v_0}\rightarrow\mathbb{R}$ that satisfy suitable asymptotic conditions, there exists a unique smooth function $\psi$ solving WE in $\mathcal{R}\cap J^+(\Sigma_{v_0})$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Penrose diagram of the exterior $\mathcal{R}$ with a constant $v$ hypersurface $\Sigma_{v_0}$.
  • Figure 2: Penrose diagram of the exterior $\mathcal{R}$ with the (shaded) region $D$.
  • Figure 3: Left: schematic representation of the spacelike foliation. Right: integration region.
  • Figure 4: The redshift estimate guarantees integrated decay in the red region (near $\mathcal{H}^+$) provided that we have integrated decay in the blue region (away from $\mathcal{H}^+$).

Theorems & Definitions (29)

  • Theorem 1.1: Well-posedness
  • Theorem 1.2: Energy boundedness
  • Theorem 1.3: Integrated decay estimate
  • Corollary 1.4: Arbitrarily fast polynomial decay
  • Proposition 2.1
  • Corollary 2.2
  • Remark 2.1
  • Lemma 2.3: Weight improvement
  • proof
  • Lemma 2.4
  • ...and 19 more