Long-time asymptotics of solutions for the second initial-boundary value problem for the damped Boussinesq equation
type: Item
instance of: scholarly article
title: Long-time asymptotics of solutions for the second initial-boundary value problem for the damped Boussinesq equation
zbMATH Open document ID: 0933.35165
DOI: 10.1155/S1085337597000407
author: V. V. Varlamov
published in: Abstract and Applied Analysis
publication date: 2000-03-29T00:00:00Z
full work available at URL: https://eudml.org/doc/48365
full work available at URL: issue-3-4.html
review text: Summary: For the damped Boussinesq equation \[ u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+ u_{xx}+\beta(u^2)_{xx},\;x\in(0,\pi),\;t>0; \] \(\alpha,b= \text{const}>0\), \(\beta=\text{const} \in\mathbb{R}^1\), the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed as a series in a small parameter present in the initial conditions, and the uniqueness of solutions is proved. The long-time asymptotics is obtained in explicit form and the blow up of the solution is examined in a certain case. The possibility of passing to the limit \(b\to+0\) in the constructed solution is investigated.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 1370656
zbMATH Keywords: long waves on the surface of shallow water
zbMATH Keywords: damped Boussinesq equation
zbMATH Keywords: small initial data
zbMATH Keywords: uniqueness
zbMATH Keywords: long-time asymptotics
zbMATH Keywords: blow up
MaRDI profile type: Publication
OpenAlex ID: W1982656688
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description: scientific article; zbMATH DE number 1370656