Existence theory for the Boussinesq equation in Modulation spaces

TL;DR

This paper establishes existence, scattering, and stability results for solutions to the generalized Boussinesq equation with initial data in modulation spaces, expanding the understanding of its behavior in these function spaces.

Contribution

It introduces a new framework for analyzing the Boussinesq equation in modulation spaces, including global/local existence and asymptotic stability results.

Findings

Existence of global and local solutions in modulation spaces.

Scattering and asymptotic stability results.

Application to specific cases like p=2, q=1, s=0.

Abstract

In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces $M^{s}_{p^\prime,q}(\mathbb{R}^n),$ $n\geq 1.$ After a decomposition of the Boussinesq equation in a $2\times 2$-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in $ M^s_{p,q}\times D^{-1}JM^s_{p,q}$ spaces for suitable $p,q$ and $s,$ including the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.