Dispersive estimates and generalized Boussinesq equation on hyperbolic spaces with rough initial data

We consider the generalized Boussinesq (GBq) equation on the real hyperbolic space \(\mathbb{H}^{n}\) (\(n\geq2\)) in a rough framework based on Lorentz spaces. First, we establish dispersive estimates for the GBq-prototype group, which is associated with a core term of the linear part of the GBq equation, through a manifold-intrinsic Fourier analysis and estimates for oscillatory integrals in \(\mathbb{H}^{n}\). Then, we obtain dispersive estimates for the GBq-prototype and Boussinesq groups on Lorentz spaces in the context of \(\mathbb{H}^{n}\). Employing those estimates, we obtain local and global well-posedness results and scattering properties in such framework. Moreover, we prove the polynomial stability of mild solutions and leverage this to improve the scattering decay.