Global existence for semi-linear hyperbolic equations in a neighbourhood of future null infinity

TL;DR

This paper proves the global existence of solutions for a class of semi-linear hyperbolic equations near future null infinity in Minkowski spacetime by employing conformal compactification and Fuchsian systems.

Contribution

It introduces a conformal compactification approach and formulates a conformal symmetric hyperbolic Fuchsian system to establish global solutions near null infinity.

Findings

Established global existence of solutions in 3+1 dimensions

Developed a conformal compactification framework

Formulated a Fuchsian system for hyperbolic equations

Abstract

In this paper, we establish the global existence of a semi-linear class of hyperbolic equations in 3+1 dimensions, that satisfy the bounded weak null condition. We propose a conformal compactification of the future directed null-cone in Minkowski spacetime, enabling us to establish the solution to the wave equation in a neighbourhood of future null infinity. Using this framework, we formulate a conformal symmetric hyperbolic Fuchsian system of equations. The existence of solutions to this Fuchsian system follows from an application of the existence theory developed in [1], and [2].