On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces

TL;DR

This paper establishes the global well-posedness of the Klein-Gordon equation with power nonlinearity for initial data in modulation spaces in dimensions three and higher, extending previous Sobolev space results.

Contribution

It proves global well-posedness in modulation spaces for a range of nonlinearities, using a high-low frequency decomposition method adapted from Bourgain's approach.

Findings

Global well-posedness for initial data in modulation spaces.

Extension of results to higher dimensions and broader nonlinearities.

Application of high-low method in a new functional setting.

Abstract

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$, in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method described by Bourgain [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.