Decay and scattering of small solutions of a generalized Boussinesq equation
type: Item
instance of: scholarly article
title: Decay and scattering of small solutions of a generalized Boussinesq equation
zbMATH Open document ID: 0884.35129
author: Yue Liu
published in: Journal of Functional Analysis
publication date: 1998-04-15T00:00:00Z
review text: Let \(u\) be a solution of the Boussinesq equation: \[ u_{tt}-u_{xx}+ \bigl( u_{xx} +f(u)\bigr)_{xx}=0 \] for \(x\in\mathbb{R}\), and denote \(u_0=u|_{t=0}\), \(v_0=v_t |_{t=0} \). Then, if \((u_0,v_0)\) is small enough in the energy norm, and if \(f(\tau)= {\mathcal O} (|\tau |^p)\) as \(\tau\to 0\) with \(p>5\), it is proven that \(u\) decays uniformly like \(|t |^{-1/3}\) when \(t\to\pm\infty\). Moreover, in this case it is also possible to construct a scattering operator for the Boussinesq equation, which acts on a neighborhood of 0 in the energy space.
reviewed by: André Martinez
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1045907
zbMATH Keywords: nonlinear scattering
zbMATH Keywords: time decay
zbMATH Keywords: Boussinesq equation
MaRDI profile type: Publication
full work available at URL: jfan.1996.3052
OpenAlex ID: W2027790027
cites work: Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation
cites work: Besov spaces in theory of approximation
cites work: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation
cites work: On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations
cites work: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation
cites work: On a class of nonlinear Schrödinger equations. II. Scattering theory, general case
cites work: Long-time behavior of solutions to nonlinear evolution equations
cites work: Global, small amplitude solutions to nonlinear evolution equations
cites work: Global existence of small solutions for a generalized Boussinesq equation
cites work: Decay and scattering of solutions of a nonlinear Schrödinger equation
cites work: Instability of solitary waves for generalized Boussinesq equations
cites work: Decay and scattering of solutions of a nonlinear relativistic wave equation
cites work: Nonlinear small data scattering for the generalized Korteweg-de Vries equation
cites work: Dispersion for non-linear relativistic equations. II
cites work: Global existence of small solutions to nonlinear evolution equations
cites work: Dispersion of low-energy waves for two conservative equations
cites work: Q3918586
cites work: Nonlinear scattering theory at low energy
DOI: 10.1006/JFAN.1996.3052
description: scientific article; zbMATH DE number 1045907