Long time existence for the semi-linear beam equation on irrational tori of dimension two

Abstract

We prove a long time existence result for the semi-linear beam equation with small and smooth initial data. We use a regularizing effect of the structure of beam equations and a very weak separation property of the spectrum of an irrational torus under a Diophantine assumption on the radius. Our approach is inspired from a paper by Zhang about the Klein–Gordon equation with a quadratic potential.