Exponential decay of the solutions of the beams system
type: Item
instance of: scholarly article
title: Exponential decay of the solutions of the beams system
zbMATH Open document ID: 0969.35130
DOI: 10.1016/S0362-546X(99)00155-8
author: Milton de Lacerda Oliveira
author: Osmundo Alves de Lima
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-10-01T00:00:00Z
review text: The paper proves the existence of the solution of the following abstract beams system: \[ u_{tt}+ A^2 u+ M(|A^{1/2}u|^2+|A^{1/2} v|^2) Au+ u_t= f,\tag{1} \] \[ v_{tt}+ A^2v+ M(|A^{1/2}u|^2+|A^{1/2} v|^2) Av+ v_t= g\tag{2} \] under the initial conditions \[ u(0)= u_0,\quad u_t(0)= u_1;\quad v(0)= v_0,\quad v_t(0)= v_1.\tag{3} \] Here, \(A\) denotes a positive selfadjoint operator of a Hilbert space with norm \(|\cdot|\), \(A^{1/2}\) is the square root of \(A\), \(M:[0,\infty)\to \mathbb{R}\) is a real function locally Lipschitz with certain properties, \(f\), \(g\) are given functions, \(u_0\), \(u_1\), \(v_0\), \(v_1\) are given constants. A weak formulation of the system (1), (2) with initial conditions (3) is presented and the pair of solutions \(u\), \(v\) is obtained as a limit of certain pairs of net functions \(u^\varepsilon\), \(v^\varepsilon\). The exponential decay of the energy \[ E(t)= \textstyle{{1\over 2}} \{|u_t|^2+|v_t|^2+|Au|^2+|Av|^2\} \] is proven.
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1529450
zbMATH Keywords: exponential decay of the energy
zbMATH Keywords: Hilbertian integral
zbMATH Keywords: existence
zbMATH Keywords: abstract beams system
zbMATH Keywords: Hilbert space
reviewed by: István Ecsedi
MaRDI profile type: Publication
cites work: Q3341259
cites work: Q5758584
cites work: Q3261641
cites work: Nonplanar oscillation of beams under periodic forcing
cites work: Stability of Nonlinear Oscillations of an Elastic Rod
cites work: Non-planar, non-linear oscillations of a beam. I: Forced motions
cites work: Q5890172
cites work: Q5549989
cites work: Mathematical analysis of the nonlinear model for the vibrations of a string
cites work: Q4720113
cites work: Q5794184
full work available at URL: s0362-546x(99)00155-8
OpenAlex ID: W2002396723
Wikidata QID: Q127674020
description: scientific article; zbMATH DE number 1529450