On The large Time Asymptotics of Klein-Gordon type equations with General Data-I

TL;DR

This paper investigates the long-time behavior of solutions to the Klein-Gordon equation with complex, space-time dependent interactions, demonstrating that solutions asymptotically resemble a free wave plus a localized component.

Contribution

It introduces a novel approach to construct the Free Channel Wave Operator for Klein-Gordon equations with general data, extending previous Schrödinger equation results to broader settings.

Findings

Global solutions asymptotically resemble a free wave and a weakly localized part.

The method applies to large, non-radial, space-time dependent data.

Generalizes previous results to arbitrary dimensions.

Abstract

We study the Klein-Gordon equation with general interaction term, which may be linear or nonlinear, and space-time dependent. The initial data is general, large and non-radial. We prove that global solutions are asymptotically given by a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works \cite{Liu-Sof1,Liu-Sof2,SW2020,SW2022}. This work generalizes the results of the first part of \cite{Liu-Sof1,Liu-Sof2} on the Schr\"odinger equation to arbitrary dimension, and non-radial data.