Scattering in the energy space for Boussinesq equations

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.

References in corpus (3)

• Kink dynamics in the ${\varphi }^4$ model: asymptotic stability for odd perturbations in the energy space
• Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation
• Nonexistence of small, odd breathers for a class of nonlinear wave equations

Cited by in corpus (4)

• The scattering problem for the $abcd$ Boussinesq system in the energy space
• Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system
• Wellposedness and scattering for the generalized Boussinesq equation
• Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region