Rafik Imekraz

Long time existence for the semi-linear Klein-Gordon equation on a compact boundaryless Riemannian manifold

ABSTRACT We investigate the long-time existence of small and smooth solutions for the semilinear Klein–Gordon equation on a compact boundary-less Riemannian manifold. Without any spectral or geometric assumption, our first result improves the lifespan obtained by the local theory. The previous result is proved under a generic condition of the mass. As a by-product of the method, we examine the particular case, where the manifold is a multidimensional torus, and we give explicit examples of algebraic masses for which we can improve the local existence time. The analytic part of the proof relies on multilinear estimates of eigenfunctions and estimates of small divisors proved by Delort–Szeftel. The algebraic part of the proof relies on a multilinear version of the Roth theorem proved by Schmidt.

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

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.