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

Jean-Marc Delort (1) , Rafik Imekraz (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 IMB - Institut de Mathématiques de Bordeaux

We investigate the long time existence of small and smooth solutions for the semi-linear Klein-Gordon equation on a compact boundaryless 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 byproduct 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 and Szeftel. The algebraic part of the proof relies on a multilinear version of the Roth theorem proved by Schmidt.

Mots clés: Klein-Gordon; compact manifold; normal form

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Marc Delort, Rafik Imekraz. Long time existence for the semi-linear Klein-Gordon equation on a compact boundaryless Riemannian manifold. Communications in Partial Differential Equations, 2017, 42 (3), pp.388-416. ⟨hal-01346519⟩