Łaba, I.; Soffer, A. Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds. (English) Zbl 0976.58019 Helv. Phys. Acta 72, No. 4, 274-294 (1999). 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. Cited in 16 Documents 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 Keywords:scattering; nonlinear Schrödinger equation; Schwarzschild manifolds; global existence; \(L^p\) estimates PDFBibTeX XMLCite \textit{I. Łaba} and \textit{A. Soffer}, Helv. Phys. Acta 72, No. 4, 274--294 (1999; Zbl 0976.58019) Full Text: arXiv