%0 Journal Article
%T Dynamic Behavior of a One-Dimensional Wave Equation with Memory and Viscous Damping
%A Jing Wang
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/984098
%X 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. 1. Introduction It is known that viscoelastic materials have been widely used in mechanics, chemical engineering, architecture, traffic, information, and so on. More and more researchers have paid close attention to the dynamic behavior and control of vibration for elastic structures with viscoelasticity over the past several decades. The most widely used models for viscoelasticity are the Boltzmann model and Kelvin-Voigt model. For instance, 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. In [4], the global existence and exponential decay of solutions of a nonlinear unidimensional wave equation with a viscoelastic boundary condition are analyzed under some assumptions. Spectral analyses of a wave equation with internal Kelvin-Voigt damping and Boltzmann damping are considered in [5, 6], respectively. The Riesz basis property of the generalized eigenfunctions of a one-dimensional hyperbolic system which is a heat equation incorporating the effect of thermomechanical coupling and the effect of inertia is studied in [7]. 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 spectrum-determined growth condition and strong exponential stability are then concluded. Similar studies from different aspects for elastic structures with viscoelasticity can also be found in [9¨C14] and the references therein. In this paper, we are
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/984098/