Multispeed Klein–Gordon Systems in Dimension Three

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 Germain [5], Germain and Masmoudi [15] and Ionescu and Pausader [4]. The main new ingredient of our method is an improved linear dispersion estimate exploiting the spherical symmetry of the Klein–Gordon flow, and a corresponding bilinear oscillatory integral estimate.