Scattering in the energy space for Boussinesq equations

TL;DR

This paper proves that small solutions to the 1D Boussinesq equation decay to zero over time within certain regions, even for supercritical nonlinearities, without parity restrictions.

Contribution

It establishes decay results for small solutions in the energy space of the generalized 1D Boussinesq equation without restrictions on nonlinearity power or initial data parity.

Findings

Small solutions decay to zero over time within specific space-time regions.

Decay holds even for supercritical nonlinearities.

No parity assumptions on initial data are required.

Abstract

In this note we show that all small solutions in the energy space of the generalized 1D Boussinesq equation must decay to zero as time tends to infinity, strongly on slightly proper subsets of the space-time light cone. Our result does not require any assumption on the power of the nonlinearity, working even for the supercritical range of scattering. No parity assumption on the initial data is needed.