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

Abstract

We consider the generalized Boussinesq (GBq) equation on the real hyperbolic space ${\mathbb{H}}^n$ ($n\ge 2$) 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.

33 pages