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

TL;DR

This paper develops dispersive estimates for the generalized Boussinesq equation on hyperbolic spaces with rough initial data, leading to new well-posedness and scattering results using Lorentz space analysis.

Contribution

It introduces dispersive estimates for the GBq equation on hyperbolic spaces in Lorentz spaces, enabling new well-posedness and scattering results for rough initial data.

Findings

Dispersive estimates established for the GBq group on hyperbolic spaces.

Proved local and global well-posedness in Lorentz spaces.

Demonstrated scattering and polynomial stability of solutions.

Abstract

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.