Space-Time resonances and the null condition for (first order) systems of wave equations

TL;DR

This paper proves global existence and asymptotic behavior of small solutions to nonlinear wave equations under a generalized null condition, extending previous null condition results.

Contribution

It introduces a generalized non-resonant condition for quadratic nonlinearities, broadening the class of wave equations with guaranteed global solutions.

Findings

Global existence of small solutions established

Linear asymptotic behavior demonstrated

Generalization of the null condition achieved

Abstract

In this manuscript we prove global existence and linear asymptotic behavior of small solutions to nonlinear wave equations. We assume that the quadratic part of the nonlinearity satisfies a non-resonant condition which is a generalization of the null condition given by Klainerman.