Wellposedness and scattering for the generalized Boussinesq equation

TL;DR

This paper establishes well-posedness, blowup, and scattering results for the generalized Boussinesq equation in various function spaces, extending understanding of its long-term behavior and solution stability.

Contribution

It proves local and global well-posedness, finite-time blowup, and scattering for the gBQ, including large radial data scattering in higher dimensions using Dodson-Murphy methods.

Findings

Local well-posedness in $L^2$ and $H^1$

Global well-posedness and scattering for small data

Finite-time blowup for certain initial conditions

Abstract

In this paper, we show the local well-posedness of the generalized Boussinesq equation(gBQ) in $L^{2}(\mathbb{R}^d), H^{1}(\mathbb{R}^d)$ and obtain the global well-posedness, finite-time blowup and small initial data scattering of gBQ in energy space $H^1(\mathbb{R}^d)$. Moreover, we obtain the large radial initial data scattering of defocusing case for $ d\geq 3 $ by using the method of Dodson-Murphy.