A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions

TL;DR

This paper proves the global existence of small solutions to quadratic nonlinear Klein-Gordon systems in two dimensions under the null condition, using algebraic methods to characterize nonlinearities.

Contribution

It provides an algebraic characterization of the null condition for Klein-Gordon systems, establishing global existence results in two dimensions.

Findings

Global existence of solutions under null condition

Solutions are asymptotically free for small initial data

Algebraic characterization of null nonlinearities

Abstract

We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, R. Xue (2004), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition.