Long time existence for the semi-linear beam equation on irrational tori of dimension two (Q2832523)

The author studies solutions of the nonlinear beam equation in two spatial dimensions equipped with periodic boundary conditions and with a positive and constant linear potential. Long-time existence result is proved for the Cauchy problem with small and smooth initial data. From a technical side, the author uses a regularizing effect of the structure of the beam equation and a very weak separation property of the spectrum of an irrational torus under a Diophantine assumption on the radius. In particular, the author incorporates methods which have been developed for the semi-linear Klein-Gordon equation on a compact Riemannian manifold with a quadratic potential by \textit{Q. Zhang} [Commun. Partial Differ. Equations 35, No. 4, 630--668 (2010; Zbl 1201.35145)].