Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space

Details Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space

2005-10-15

arxiv.org/abs/math/0510315v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

10.1007/s00220-006-0101-6

We provide a uniform decay estimate of Morawetz type for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. This estimate is both uniform in space and time, so in particular it implies a uniform bound on the sup norm of solutions which can be given in terms of certain inverse powers of the radial and advanced/retarded time coordinate variables. As a model application, we show these estimates give a very simple proof small amplitude scattering for nonlinear scalar fields with higher than cubic interactions.

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