The global existence of small-amplitude solutions to nonlinear Klein-Gordon equations: A study based on S. Klainerman's approach

TL;DR

This paper analyzes Klainerman's proof of the global existence and decay of small-amplitude solutions to nonlinear Klein-Gordon equations in four dimensions, using advanced Sobolev and energy methods.

Contribution

It provides a detailed study and validation of Klainerman's approach, extending understanding of global solutions for small initial data in nonlinear Klein-Gordon equations.

Findings

Solutions exist globally for small initial data

Solutions decay at a rate of t^{-5/4}

Analytical techniques include Sobolev norms and energy estimates

Abstract

In this thesis we explore S. Klainerman's proof on the global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, as established in his paper from 1985. We consider initial data with small amplitude and compact support and aim prove the global existence and uniform decay of smooth solutions. We establish that solutions exist globally if the initial data satisfy a suitable smallness condition. Key analytical tools include generalized Sobolev norms and uniform decay estimates for the associated linear problem. The solutions exhibit a decay rate of $t^{-5/4}$, uniform in time and space. This result is achieved by combining the energy method, perturbed Klein-Gordon techniques, and Sobolev inequalities.