A Fuchsian viewpoint on the weak null condition

TL;DR

This paper proves the existence of solutions for certain semilinear wave equations in 3+1 dimensions under a special bounded weak null condition, using the Fuchsian method to transform and solve the equations near spatial infinity.

Contribution

It introduces the bounded weak null condition and applies the Fuchsian method to establish solution existence for wave systems near spatial infinity.

Findings

Solutions exist near spatial infinity for small initial data.

The Fuchsian method effectively transforms and solves the wave equations.

The approach extends the applicability of the weak null condition analysis.

Abstract

We analyze systems of semilinear wave equations in $3+1$ dimensions whose associated asymptotic equation admit bounded solutions for suitably small choices of initial data. Under this special case of the weak null condition, which we refer to as the \textit{bounded weak null condition}, we prove the existence of solutions to these systems of wave equations on neighborhoods of spatial infinity under a small initial data assumption. Existence is established using the Fuchsian method. This method involves transforming the wave equations into a Fuchsian equation defined on a bounded spacetime region. The existence of solutions to the Fuchsian equation then follows from an application of the existence theory developed in \cite{BOOS:2020}. This, in turn, yields, by construction, solutions to the original system of wave equations on a neighborhood of spatial infinity.