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

P. Blue·A. Soffer
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Abstract

We consider the wave equation (-\dt^2+\dr^2 -V -V_L(-Δ_{S^2})) u = fF'(|u| ^2) u with (t,ρ,θ,ϕ) in R x R x 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_{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.

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P. Blue, A. Soffer. 2007-03-13. A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold. https://doi.org/10.1007/s11005-007-0177-8

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