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We consider the wave equation (-\dt^2+\dr^2 -V -V_L(-\Delta_{S^2})) u =fF'(|u| ^2) u with (t,\rho,\theta,\phi) in R x R x S^2. The wave equation on aspherically symmetric manifold with a single closed geodesic surface or on theexterior of the Schwarzschild manifold can be reduced to this form. Using asmoothed Morawetz estimate which does not require a spherical harmonicdecomposition, we show that there is decay in L^2_{loc} for initial data in theenergy class, even if the initial data is large. This requires certainconditions on the potentials V, V_L, and f. We show that a key condition on theweight in the smoothed Morawetz estimate can be reduced to an ODE condition,which is verified numerically.Comment: 10 pages, 4 figure

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