A large data theory for nonlinear wave on the Schwarzschild background. (English) Zbl 1507.83052
Summary: We study both of the scattering and Cauchy problems for the semilinear wave equation with a null quadratic form on the Schwarzschild background. Prescribing the scattering data that are given by the short pulse data on the future null infinity and are trivial on the future event horizon, we construct a class of global solutions backwards up to any finite time and show that the wave travels in such a way that almost all of the (large) energy is focusing in an outgoing null strip, while little radiates out of this strip. In reverse, considering a class of Cauchy data with large energy norms, there exists a unique and global solution in the future development, and the radiation field along the future null infinity exists. Furthermore, most of the wave packet is confined in an incoming null strip and reflected to the future event horizon, whereas little is transmitted to the future null infinity.
MSC:
| 83C57 | Black holes |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35L05 | Wave equation |
| 15A63 | Quadratic and bilinear forms, inner products |
| 81U05 | \(2\)-body potential quantum scattering theory |
| 83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 83C10 | Equations of motion in general relativity and gravitational theory |
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