Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions

TL;DR

This paper introduces a weaker condition than the weak null condition for ensuring global existence of small solutions to semilinear wave systems in three dimensions and analyzes their asymptotic behavior.

Contribution

It proposes a new, less restrictive condition for global existence and studies the asymptotic pointwise behavior of solutions under this condition.

Findings

Established a weaker sufficient condition for global existence.

Derived the asymptotic behavior of solutions under the new condition.

Extended results to solutions satisfying Alinhac's original condition.

Abstract

In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.