# A large data theory for nonlinear wave on the Schwarzschild background

## TL;DR

This paper develops a comprehensive theory for nonlinear wave behavior on Schwarzschild black hole backgrounds, analyzing scattering and Cauchy problems, and revealing how energy concentrates or disperses in different spacetime regions.

## Contribution

It introduces a new large data theory for nonlinear waves on Schwarzschild spacetime, detailing the energy focusing and reflection phenomena for scattering and Cauchy problems.

## Key findings

- Most energy concentrates in outgoing null strips during scattering.
- Large energy Cauchy data lead to globally smooth solutions with wave reflection.
- Wave packets are confined or transmitted depending on initial data configuration.

## Abstract

We study both of the scattering and Cauchy problems for the semilinear wave equation with 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 globally smooth 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 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.