Abstract
We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior \(|\phi| = O\left(r^{-1}\big|t-|r^*|\big|^{-\frac{1}{2}}\right)\) as long as the source term is bounded in the norm \((1-\frac{2M}{r})^{-1}\cdot(1 + t + |r^*|)^{-1}L^1\big(H^3_\Omega(r^2dr^* d\omega)\big)\). In particular this gives scattering at small amplitudes for non-linear scalar fields of the form \(\square_{g}\varphi = \lambda |\varphi |^{p}\varphi \) for all 2 < p.
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Communicated by P. Constantin
This paper is dedicated to the memory of Hope Machedon
The second author would like thank MSRI and Princeton University, where a portion of this research was conducted during the Fall of 2005. The second author was also supported by a NSF postdoctoral fellowship.
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Blue, P., Sterbenz, J. Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space. Commun. Math. Phys. 268, 481–504 (2006). https://doi.org/10.1007/s00220-006-0101-6
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DOI: https://doi.org/10.1007/s00220-006-0101-6