## Abstract

We consider the wave equation (-∂_{t}^{2} + ∂_{ρ}^{2} - V - V_L(-Δ_{S2}))u = f F'(|u|^{2})u with (t, ρ, θ, φ) in ℝ × ℝ × S^{2}. 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 L_{loc}^{2} 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 language | English (US) |
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Pages (from-to) | 227-238 |

Number of pages | 12 |

Journal | Letters in Mathematical Physics |

Volume | 81 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2007 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

## Keywords

- Local decay estimates
- Schwarzschild manifold