General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history
type: Item
instance of: scholarly article
title: General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history
zbMATH Open document ID: 1403.35043
author: Baowei Feng
published in: Mediterranean Journal of Mathematics
publication date: 2018-07-24T00:00:00Z
review text: The author investigates the initial-boundary value problem for a viscoelastic wave equation with dynamic boundary conditions, source term and a nonlinear weak damping realized on a part of the boundary and a past history: \[ \begin{aligned} &u_{tt}(t)-\Delta(u+u_t)(t)+\int_0^\infty g(s)\Delta u(t-s)\,\text{d}s=0\;\text{ in }\Omega\times \mathbb{R}^+, \;u(t)=0\;\text{ on }\Gamma_0\times \mathbb{R}^+,\\ &u_{tt}(t)= - \frac{\partial (u+u_t)}{\partial \nu}+\int_0^\infty g(s)\frac{\partial u}{\partial \nu}(t-s)\,\text{d}s-h(u_t)-f(u)\;\text{ on } \Gamma_1\times \mathbb{R}^+, \\ &u(x,-t)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega, \end{aligned} \] where \(\Omega\subset \mathbb{R}^n (n\geq 1)\) is a regular and bounded domain, \(\partial \Omega =\Gamma_0\cup \Gamma_1,\;\Gamma_0\cap \Gamma_1=\emptyset.\) Using a new variable \(\eta(x,t,s)=u(t)-u(t-s)\) and assuming \(1-\int _0^\infty = \ell>0\) the problem is transformed to \[ \begin{aligned} &u_{tt}(t)-\ell\Delta u(t)-\Delta u_t(t)-\int_0^\infty g(s)\Delta \eta(s)\,\text{d}s=0\;\text{ in }\Omega\times \mathbb{R}^+, \;u(t)=0\;\text{ on } \Gamma_0\times \mathbb{R}^+,\\ &u_{tt}(t)= - \ell\frac{\partial u}{\partial \nu}-\frac{\partial u_t}{\partial \nu}-\int_0^\infty g(s)\frac{\partial \eta}{\partial \nu}(s)\,\text{d}s-h(u_t)-f(u)\;\text{ on } \Gamma_1\times \mathbb{R}^+, \\ &u(x,-t)=u_0(x),\;u_t(x,0)=u_1(x),\;\eta_0=u_0(x)-u_0(x,-s),\;x\in \Omega. \end{aligned} \] The main result is the general decay of energy to the transformed problem.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 74D05
zbMATH DE Number: 6909942
zbMATH Keywords: convexity
zbMATH Keywords: boundary feedback
zbMATH Keywords: infinite memory
zbMATH Keywords: nonlinear weak boundary damping
MaRDI profile type: Publication
full work available at URL: s00009-018-1154-4
OpenAlex ID: W2800216794
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Wikidata QID: Q129858808
DOI: 10.1007/S00009-018-1154-4
description: scientific article; zbMATH DE number 6909942