Existence of strong global attractors for hyperbolic equation with linear memory
type: Item
instance of: scholarly article
title: Existence of strong global attractors for hyperbolic equation with linear memory
zbMATH Open document ID: 1068.45018
author: Qiaozhen Ma
author: Chengkui Zhong
published in: Applied Mathematics and Computation
publication date: 2004-11-23T00:00:00Z
review text: The authors consider the following nonlinear integro-differential equation which describes a homogeneous and isotropic viscoelastic solid: \[ \begin{cases} u_{tt}+ \alpha u_t - K(0)\Delta u -\int^\infty_0 {K^ \prime}(s) \Delta u(t-s)ds + g(u)= f,&\text{in }\Omega\times \mathbb{R}^+,\\ u (x,t) = 0, &x \in \Gamma,\;\;t\in{\mathbb{R}}, \\ u (x,t) = u_0 (x,t), &x\in \Omega,\;\;t\leq 0, \end{cases}\tag{1} \] where \(\Omega\) is an open and bounded subset of \({\mathbb {R}}^n \) with smooth boundary \( \Gamma.\) \( \alpha > 0\) is a given constant; and \(K^\prime (s) \leq 0 \) for all \(s\in {\mathbb{R}^+},\) and \(K(0), K(\infty)\) are positive. \(g\in C^2 ({\mathbb {R}}),\) and \(f\in H^1_ 0 (\Omega). \) Under suitable conditions, and using semigroup theory, the authors prove the existence of a global attractor for problem (1).
reviewed by: Mohamed O. El-Doma
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 2117761
zbMATH Keywords: global attractor
zbMATH Keywords: linear memory
zbMATH Keywords: hyperbolic equation
zbMATH Keywords: \(\omega\)-limit compact semigroup
zbMATH Keywords: nonlinear integro-differential equation
zbMATH Keywords: isotropic viscoelastic solid
MaRDI profile type: Publication
full work available at URL: j.amc.2003.08.080
OpenAlex ID: W2056518170
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DOI: 10.1016/J.AMC.2003.08.080
description: scientific article; zbMATH DE number 2117761