Long-time existence for semi-linear Klein-Gordon equations with quadratic potential

TL;DR

This paper proves extended existence times for small smooth solutions to semi-linear Klein-Gordon equations with quadratic potential, using normal form methods to handle spectral challenges.

Contribution

It extends the lifespan of solutions beyond local existence for almost all masses by employing normal form techniques on the Sobolev energy.

Findings

Solutions exist longer than local theory predicts

Normal form method effectively manages spectral eigenvalue issues

Results apply to almost every mass value

Abstract

We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the Sobolev energy. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvalues $\lambda, \lambda'$ of $\sqrt{-\Delta+|x|^2}$ may be separated by a distance as small as $\frac{1}{\sqrt{\lambda}}$.