The zero mass problem for Klein-Gordon equations

TL;DR

This paper establishes global solutions and decay results for Klein-Gordon equations with possibly vanishing mass, using novel techniques combining flat and hyperboloidal foliations of Minkowski spacetime.

Contribution

It provides the first rigorous proof of unified decay results for Klein-Gordon equations with zero or small mass, employing a new iteration method that combines different spacetime foliations.

Findings

Proved global existence for Klein-Gordon equations with vanishing mass.

Established unified decay estimates using combined foliations.

Introduced an iteration procedure to leverage multiple spacetime foliations.

Abstract

We are interested in the global solutions to a class of Klein-Gordon equations, and particularly in the unified time decay results with respect to the possibly vanishing mass parameter. We give for the first time a rigorous proof, which relies on both the flat foliation and the hyperboloidal foliation of the Minkowski spacetime. In order to take advantages of both foliations, an iteration procedure is used.