Wave and Klein-Gordon equations on hyperbolic spaces

Jean-Philippe Anker (1) , Vittoria Pierfelice (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We consider the Klein-Gordon equation associated with the Laplace-Beltrami operator Δ on real hyperbolic spaces of dimension n≥2; as Δ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well-posedness results for the corresponding semilinear equation with low regularity data.

Mots clés: Hyperbolic space; wave kernel; semilinear wave equation; semilinear Klein-Gordon equation; dispersive estimate; Strichartz estimate; global well-posedness

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]

Jean-Philippe Anker, Vittoria Pierfelice. Wave and Klein-Gordon equations on hyperbolic spaces. 2011. ⟨hal-00581773v1⟩