Global solutions of coupled Klein-Gordon equations with different velocities in four space-time dimensions

TL;DR

This paper establishes global solutions for coupled nonlinear Klein-Gordon equations with different velocities and masses in four dimensions, using advanced energy estimates and bootstrap methods.

Contribution

It introduces a novel combination of classical bootstrap, conformal energy estimates, and Hardy inequalities to prove global existence for this class of coupled equations.

Findings

Global solutions are proven to exist under broad quadratic nonlinearities.

The method effectively handles different velocities and masses in the coupled system.

The approach extends previous techniques to more complex coupled Klein-Gordon systems.

Abstract

In this article one will discuss the system of coupled nonlinear Klein-Gordon equations with different velocities and different masses. The nonlinearity considered is a general quadratic nonlinearity without any restriction. The method is a classical boot-strap argument combined with a serious of techniques including conformal energy estimate, global sobolev's lemma and Hardy type inequalities.