---
abstract: 'We consider long time evolution of small solutions to general multispeed Klein-Gordon systems on $\mathbb{R}\times\mathbb{R}^{3}$. We prove that such solutions are always global and scatter to a linear flow, thus extending partial results obtained previously in [@G11; @GM11; @IP12]. The main new ingredients of our method is an improved *linear* dispersion estimate exploiting the asymptotic spherical symmetry of Klein-Gordon waves, and a corresponding *bilinear* oscillatory integral estimate.'
address: 'Courant Institute of Mathematical Sciences, New York University, New York, NY 10012'
author:
- Yu Deng
title: 'Multispeed Klein-Gordon systems in dimension three'
---

[^1]

Introduction {#intro3}
============

In this paper we consider a system of quasilinear, multispeed Klein-Gordon equations in space dimension three, namely $$\label{nlkg}(\partial_{t}^{2}-c_{\alpha}^{2}\Delta+b_{\alpha}^{2})u_{\alpha}=\mathcal{Q}_{\alpha}(u,\partial u,\partial^{2}u),\qquad 1\leq\alpha\leq d,$$ where the speeds $c_{\alpha}$ and masses $b_{\alpha}$ are arbitrary positive numbers, and $\mathcal{Q}$ is a quadratic, quasilinear nonlinearity satisfying a suitable symmetry condition.

We are mostly interested in the behavior of small solutions to (\[nlkg\]), which has been studied in many previous works. A typical phenomenon that happens for such solutions is global regularity and scattering, which is suggested by the dispersive nature of linear Klein-Gordon equation; to justify this one needs to combine the standard energy estimate with a suitable 