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Letters in Mathematical Physics

, Volume 81, Issue 3, pp 227–238 | Cite as

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

  • Pieter Blue
  • Avy Soffer
Article

Abstract

We consider the wave equation \((-\partial_t^{2} + \partial_\rho^{2} - V - V_L(-\Delta_{S^2}))u = f F'(|u|^2)u\) with \((t, \rho, \theta, \phi)\) in \(\mathbb R \times \mathbb R \times S^2\) . 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 \(L^2_{\rm {loc}}\) 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.

Mathematics Subject Classifications (2000)

35P25 58-xx 

Keywords

Schwarzschild manifold local decay estimates. 

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References

  1. 1.
    Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. math.AP/ 0511281Google Scholar
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    Blue, P., Soffer, A.: Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole. math.AP/0612168Google Scholar
  3. 3.
    Blue P. and Sterbenz J. (2006). Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2): 481–504 zbMATHCrossRefADSGoogle Scholar
  4. 4.
    Dafermos, M., Rodnianski, I.: The red-shift effect and radiation decay on black hole spacetimes. gr-qc/0512119Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of Mathematics, Hill Center-Busch Campus, RutgersThe State University of New JerseyPiscatawayUSA

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