Strong instability of solitary-wave solutions of a generalized Boussinesq equation
type: Item
instance of: scholarly article
title: Strong instability of solitary-wave solutions of a generalized Boussinesq equation
zbMATH Open document ID: 0973.35163
author: Yue Liu
published in: Journal of Differential Equations
publication date: 2000-10-10T00:00:00Z
review text: The author examines a generalized Boussinesq equation \[ u_{tt}- u_{xx}+ (u_{xx}+ f(u))_{xx}= 0,\quad x\in\mathbb{R},\quad t>0 \] with \(f\in C^1(\mathbb{R})\). This model arises by describing water waves, phase transitions for shape memory alloys, and in the theory of anharmonic lattice waves. It is known that the above equation possesses travelling-wave type solutions with finite energy, which are called here solitary-wave solutions. First, by constructing a special functional of the solution, the author proves strong instability of solitary-wave solutions. Then some improved blow-up results are established in the particular case \(f(u)=|u|^{p-1} u\), \(p>1\).
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 37K45
Mathematics Subject Classification ID: 76B25
zbMATH DE Number: 1515442
zbMATH Keywords: travelling-wave solutions
zbMATH Keywords: generalized Boussinesq equation
zbMATH Keywords: water waves
zbMATH Keywords: phase transitions
zbMATH Keywords: shape memory alloys
zbMATH Keywords: anharmonic lattice waves
zbMATH Keywords: solitary-wave solutions
zbMATH Keywords: functional
zbMATH Keywords: strong instability
zbMATH Keywords: blow-up
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3765
OpenAlex ID: W2047573770
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DOI: 10.1006/JDEQ.2000.3765
description: scientific article; zbMATH DE number 1515442