Wave and Klein-Gordon equations on hyperbolic spaces

TL;DR

This paper analyzes wave and Klein-Gordon equations on hyperbolic spaces, deriving dispersive and Strichartz estimates, and establishing global well-posedness for semilinear equations with low regularity data.

Contribution

It provides new dispersive and Strichartz estimates for Klein-Gordon and wave equations on hyperbolic spaces, enabling low-regularity global well-posedness results.

Findings

Derived dispersive estimates for Klein-Gordon and wave equations on hyperbolic spaces.

Established Strichartz estimates for a broad class of admissible pairs.

Proved global well-posedness for semilinear equations with low regularity initial data.

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.