Well-posedness and scattering for wave equations on hyperbolic spaces with singular data

Lucas C. F. Ferreira, Pham Truong Xuan

Published 2023-03-11, updated 2024-07-16Version 2

We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on 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.