Well-Posedness and Scattering for Wave Equations on Hyperbolic Spaces With Singular Data

Abstract

We consider wave equations with critical shift on the real hyperbolic space ${\mathbb{H}}^n$ ( $n\ge 2$ ) in the framework of weak- $L^p$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of ${\mathbb{H}}^n$ . Then, employing those estimates, we prove global well-posedness of solutions and an exponential asymptotic stability property. Moreover, we develop a scattering theory and construct wave operators in such singular framework.