A note on a system of cubic nonlinear Klein-Gordon equations in one space dimension

TL;DR

This paper investigates the global existence and decay properties of solutions to a system of cubic nonlinear Klein-Gordon equations in one dimension, removing the need for compact support assumptions on initial data.

Contribution

It establishes global solutions and decay rates under a structural condition, extending previous results that required compact support of initial data.

Findings

Solutions exist globally in time

Solutions decay at a rate of O(t^{-1/2}) in L^ty

No compact support condition on initial data is needed

Abstract

We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order $O(t^{-1/2})$ in $L^\infty$ as $t$ tends to infinity without the condition of a compact support on the Cauchy data which was assumed in the previous works.