Table of Contents
ISRN Applied Mathematics
Volume 2014 (2014), Article ID 984098, 12 pages
http://dx.doi.org/10.1155/2014/984098
Research Article

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

School of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 9 January 2014; Accepted 17 March 2014; Published 1 April 2014

Academic Editors: Y. Dimakopoulos, M. Hermann, and M.-H. Hsu

Copyright © 2014 Jing Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.