Global existence of small amplitude solution to nonlinear system of wave and Klein-Gordon equations in four space-time dimensions

TL;DR

This paper introduces a hyperboloidal foliation energy method that unifies the treatment of wave and Klein-Gordon equations, establishing global solutions for small amplitude coupled systems in four dimensions.

Contribution

It develops a new hyperboloidal foliation method applicable to coupled wave and Klein-Gordon equations, allowing for broader nonlinear term inclusion and global existence results.

Findings

Established global existence of small solutions in 4D for coupled wave and Klein-Gordon equations.

The method includes important nonlinear terms not covered by previous approaches.

Unified treatment of wave and Klein-Gordon equations using hyperboloidal foliation.

Abstract

In this article one will develop a so-called hyperboloidal foliation method, which is an energy method based on a foliation of space-time into hyperboloidal hypersurfaces. This method permits to treat the wave equations and the Klein-Gordon equations in the same framework so that one can apply it to the coupled systems of wave and Klein-Gordon equations. As an application, one will establish the global-in-time existence of small amplitude solution to the coupled wave and Klei-Gordon equations with quadratic nonlinearity in four space-time dimensions under certain conditions. Compared with those introduced by S. Katayama, the conditions imposed in this article permit to include some important nonlinear terms. All of these suggests that this method may be a more natural way of regarding the wave operator.