Exponential stability of a one-dimensional thermoviscoelastic system with memory type

In this paper, we study the stability for a one-dimensional linear thermoviscoelastic equation with memory type for Dirichlet-Dirichlet boundary conditions. A detailed spectral analysis gives that the spectrum of the system contains two parts: the point and continuous spectrum. It is shown that there are three classes of eigenvalues: one is along the negative real axis approaching to -∞, the second is approaching to a vertical line which parallels to the imagine axis, and the third class is distributed around the continuous spectrum which are accumulation points of the last classes of eigenvalues. Moreover, it is pointed out that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition holds true and the exponential stability of the system is then established.