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

2024/10/27 by Lucas C. F. Ferreira, Ferreira, Lucas C. F., Pham Truong Xuan +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #advanced mathematical theories #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2410.20472

Abstract

We consider the generalized Boussinesq (GBq) equation on the real hyperbolic space ℍⁿ (n≥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 ℍⁿ. Then, we obtain dispersive estimates for the GBq-prototype and Boussinesq groups on Lorentz spaces in the context of ℍⁿ. 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.

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