Dynamic Behavior of a One-Dimensional Wave Equation with Memory and Viscous Damping

Jing Wang

doi:10.1155/2014/984098

Abstract

We study the dynamic behavior of a one-dimensional wave equation with both exponential polynomial kernel memory and viscous damping under the Dirichlet boundary condition. By introducing some new variables, the time-variant system is changed into a time-invariant one. The detailed spectral analysis is presented. It is shown that all eigenvalues of the system approach a line that is parallel to the imaginary axis. The residual and continuous spectral sets are shown to be empty. The main result is the spectrum-determined growth condition that is one of the most difficult problems for infinite-dimensional systems. Consequently, an exponential stability is concluded.

Highlights

It is known that viscoelastic materials have been widely used in mechanics, chemical engineering, architecture, traffic, information, and so on
The results concerning the exponential asymptotic stability of a linear hyperbolic integrodifferential equation in Hilbert space are established in [1], which is an abstract version of the equation of motion for dynamic linear viscoelastic solids
In [2, 3], the exponential stabilities of a vibrating beam with one segment made of viscoelastic material of a Kelvin-Voigt type and a vibrating string with Boltzmann damping are proved under certain hypotheses of the smoothness and structural condition of the coefficients of the system

Summary

Introduction

It is known that viscoelastic materials have been widely used in mechanics, chemical engineering, architecture, traffic, information, and so on. In [2, 3], the exponential stabilities of a vibrating beam with one segment made of viscoelastic material of a Kelvin-Voigt type and a vibrating string with Boltzmann damping are proved under certain hypotheses of the smoothness and structural condition of the coefficients of the system. In [8], a detailed spectral analysis for a heat equation system which is derived from a thermoelastic equation with memory type is presented and the spectrumdetermined growth condition and strong exponential stability are concluded. We are interested in the following onedimensional wave equation with viscoelastic damping under the Dirichlet boundary condition:. The main result is the spectrum-determined growth condition that is presented in Section 4; a strongly exponential stability is obtained

System Operator Setup

Spectral Analysis of System Operator

Spectrum-Determined Growth Condition and Exponential Stability