A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold

Pieter Blue, Avy Soffer

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

We consider the wave equation (-∂t2 + ∂ρ2 - V - V_L(-ΔS2))u = f F'(|u|2)u with (t, ρ, θ, φ) in ℝ × ℝ × S2. The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in Lloc2 for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V L and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.

Original languageEnglish (US)
Pages (from-to)227-238
Number of pages12
JournalLetters in Mathematical Physics
Volume81
Issue number3
DOIs
StatePublished - Sep 2007

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Local decay estimates
  • Schwarzschild manifold

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