A nonlinear beam equation arising in the theory of elastic bodies
type: Item
instance of: scholarly article
title: A nonlinear beam equation arising in the theory of elastic bodies
zbMATH Open document ID: 0902.35066
DOI: 10.4171/ZAA/798
author: K. Doppel
author: K. Pflüger
author: Wolfgang Herfort
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-09-09T00:00:00Z
review text: The authors study a nonlinear Cauchy problem which arises in the theory of oscillations in elastic bodies under external forces. More precisely, they consider the problem \[ \partial_{t}^{2}w + \partial_{\xi}^{4}w + \gamma_{1}\partial_{t} \partial_{\xi}^{4}w - \gamma_{2}\partial_{\xi}^{2}w - \gamma_{3} \left ( \int_{0}^{1} (\partial_{\xi}w)^{2} \;d \xi \right) \partial_{\xi}^{2}w - \gamma_{4}\partial_{t}w - \gamma_{5}(\partial_{t}w)^{3} = 0, \] \[ w(\xi,0)= w_{0}(\xi), \quad \partial_{t}w(\xi,0) = w_{1}(\xi),\quad w(0,t) = w(1,t) = \partial_{\xi}^{2}w(0,t) = \partial_{\xi}^{2} w(1,t) = 0,\tag{1} \] where \((\xi,t)\in(0,1)\times[0,+\infty)\), \(\gamma_{1},...,\gamma_{5}\) are real parameters, \(\gamma_{1} \geq 0.\) The type of equation which appears in (1) has been suggested by Blevins (\textit{R. D. Blevins}, Flow-induced vibrations (Van Nostrand Reinhold Comp., New York 1977; Zbl 0385.73001)]. The initial conditions in (1) are usual, whereas the boundary one (which describe the so-called simply supported ends), have been proposed by some studies in engineering [see \textit{W. Herfort} and \textit{H. Troger}, Robust modelling of flow induced oscillations of bluff bodies. Math. Modelling 8, 251-255 (1987)]. These boundary conditions cause some additional difficulties, due to the fact that classical Sobolev spaces can not be used. The authors show the global solvability of (1). To prove this, they reformulate the problem as a nonlinear Cauchy problem in an appropriate Sobolev space which contains the boundary conditions. Then, they study the associated linear problem (by using strongly continuous contraction semigroups) and obtain some estimates for the nonlinear part. Next, this Cauchy problem is transformed into an integral equation which gives rise to a fixed point problem. By applying some previous results [see \textit{C. Foiaş, G. Gussi}, and \textit{V. Poenaru}, Rev. Roum. Math. Pures Appl. 3, 283-304 (1958; Zbl 0087.11604)], it is proved the existence and uniqu…
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 1129178
zbMATH Keywords: nonlinear beam equations
zbMATH Keywords: initial and boundary value problems
zbMATH Keywords: global solvability
zbMATH Keywords: oscillation in elastic bodies
zbMATH Keywords: contraction semigroups
zbMATH Keywords: fixed point methods
reviewed by: Antonio Cañada
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/798
OpenAlex ID: W2034050344
description: scientific article; zbMATH DE number 1129178