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Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds. (English) Zbl 0976.58019

Summary: We consider the nonlinear Schrödinger equation with a pure power repulsive nonlinearity on Schwarzschild manifolds. Equations of this type arise when a nonlinear wave equation on a Schwarzschild manifold is written in Hamiltonian form. For radial solutions with sufficiently localized initial data, we obtain global existence, \(L^p\) estimates, and the existence and asymptotic completeness of the wave operators. Our approach is based on a dilation identity and global space-time estimates.

MSC:

58J45 Hyperbolic equations on manifolds
35Q75 PDEs in connection with relativity and gravitational theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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