Stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation
type: Item
instance of: scholarly article
title: Stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation
zbMATH Open document ID: 1195.35228
author: Tae Gab Ha
author: Jong Yeoul Park
published in: Dynamics of Partial Differential Equations
publication date: 2010-02-08T00:00:00Z
review text: The authors consider the viscoelastic Euler-Bernoulli type initial-boundary value problem with a localized damping term: \[ u_{tt}+\Delta^2u-M(\| u\|^2)\Delta u- \int_0^tg(t-\tau)\Delta^2u(\tau)\,d\tau+\rho(x,u_t)=0\quad \text{in }\Omega\times \mathbb{R}_+, \] \[ u=\frac {\partial u}{\partial \nu}=0\quad \text{on }\Gamma\times \mathbb{R}_+; \qquad u(x,0)=u_0(x), \quad u'(x,0)=u_1(x),\qquad x\in \Omega, \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(n\geq 1\), with a boundary \(\Gamma=\Gamma_0\cup \Gamma_1\) of the \(C^2\) class, where \(\Gamma_0\) and \(\Gamma_1\) are closed and disjoint, \(\Gamma_0\neq \emptyset\) and satisfying the conditions \( m\cdot \nu\geq 0>0\) on \(\Gamma_1\), \(m\cdot \nu\leq 0\) on \(\Gamma_0\), \(m(x)=x-x_0\) \((x_0\in \mathbb{R}^n)\). Further \(M\in C^1(\mathbb{R}_+)\), \(g:\mathbb{R}_+\to \mathbb{R}_+\) and \(\rho(x,s)\) is almost everywhere differentiable in \(x\) and nondecreasing in \(s\). The work is devoted to prove the existence of global solutions and decay for the energy of solutions. Due to my opinion the assumptions for the boundary cannot be fulfilled for the \(C^2\) boundary. The strong convergence (3.22) is wrong, in order to pass to the limit in the equation it suffices the strong convergence in \(C([0,T];H^1_0(\Omega))\) instead of \(C([0,T];V)\), \(V=H^2_0(\Omega)\).
Mathematics Subject Classification ID: 35L76
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74Dxx
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 5667598
zbMATH Keywords: existence of solution
zbMATH Keywords: energy decay
zbMATH Keywords: Euler-Bernoulli equation
zbMATH Keywords: local dissipation
zbMATH Keywords: localized damping term
reviewed by: Igor Bock
MaRDI profile type: Publication
Wikidata QID: Q115204761
DOI: 10.4310/DPDE.2009.V6.N4.A3
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description: scientific article; zbMATH DE number 5667598