Vol. 7, No. 4, 2014

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Wave and Klein–Gordon equations on hyperbolic spaces

Jean-Philippe Anker and Vittoria Pierfelice

Vol. 7 (2014), No. 4, 953–995
Abstract

We consider the Klein–Gordon equation associated with the Laplace–Beltrami operator $\Delta$ on real hyperbolic spaces of dimension $n\ge 2$; as $\Delta$ 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.

Keywords
hyperbolic space, wave kernel, semilinear wave equation, semilinear Klein–Gordon equation, dispersive estimate, Strichartz estimate, global well-posedness
Mathematical Subject Classification 2010
Primary: 35L05, 43A85, 43A90, 47J35
Secondary: 22E30, 35L71, 58D25, 58J45, 81Q05