Global existence of small amplitude solutions to one-dimensional nonlinear Klein-Gordon systems with different masses

TL;DR

This paper proves that small, smooth, and compactly supported initial data lead to global solutions that decay over time for a class of one-dimensional nonlinear Klein-Gordon systems with different masses, including cases with mass resonance.

Contribution

It establishes the global existence and decay rates of solutions to cubic nonlinear Klein-Gordon systems with different masses under a structural condition, even in mass resonance cases.

Findings

Solutions exist globally for small initial data.

Decay rate of solutions is $O(t^{-(1/2-1/p)})$ in $L^p$ norms.

Results hold even in the presence of mass resonance.

Abstract

We study the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the rate $O(t^{-(1/2-1/p)})$ in $L^p$, $p\in[2,\infty]$ as $t$ tends to infinity even in the case of mass resonance, if the Cauchy data are sufficiently small, smooth and compactly supported.