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

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 thesystem 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 isone of the most difficult problems for infinite-dimensional systems. Consequently, an exponential stability is concluded.