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- Title
Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space.
- Authors
Blue, Pieter; Sterbenz, Jacob
- 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.
- Subjects
NONLINEAR wave equations; SCHWARTZ spaces; DECAY schemes (Radioactivity); ASYMPTOTIC theory in mathematical physics; ASYMPTOTIC theory in nonlinear differential equations; SCALAR field theory
- Publication
Communications in Mathematical Physics, 2006, Vol 268, Issue 2, p481
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-006-0101-6