Blue, Pieter; Soffer, Avy A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold. (English) Zbl 1137.58011 Lett. Math. Phys. 81, No. 3, 227-238 (2007). Summary: 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 \(L^2_{\text{loc}}\) 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. Cited in 13 Documents MSC: 58J45 Hyperbolic equations on manifolds 35P25 Scattering theory for PDEs 35L70 Second-order nonlinear hyperbolic equations Keywords:Schwarzschild manifold; local decay estimates PDFBibTeX XMLCite \textit{P. Blue} and \textit{A. Soffer}, Lett. Math. Phys. 81, No. 3, 227--238 (2007; Zbl 1137.58011) Full Text: DOI arXiv References: [1] Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. math.AP/ 0511281 · Zbl 1158.83007 [2] Blue, P., Soffer, A.: Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole. math.AP/0612168 · Zbl 1076.58020 [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 · Zbl 1123.58018 [4] Dafermos, M., Rodnianski, I.: The red-shift effect and radiation decay on black hole spacetimes. gr-qc/0512119 · Zbl 1169.83008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.