Global existence of null-form wave equations on small asymptotically Euclidean manifolds

TL;DR

This paper proves the global existence of small solutions to null-form wave equations on asymptotically Euclidean manifolds with small metric perturbations, extending results to certain non-null systems.

Contribution

It establishes global existence results for null-form wave equations on manifolds with metrics close to Euclidean, including small perturbations and systems without null conditions.

Findings

Global existence of small solutions on asymptotically Euclidean manifolds.

Extension to systems without null condition under small perturbations.

Analysis of solutions with decay properties of the metric.

Abstract

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on $(R^3, g)$, where the metric $g$ is a small perturbation of the flat metric and approaches the Euclidean metric like $(1+|x|)^{-a}$ with $a>1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in the appendix.