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

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## 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.## Preview

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## References

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