Dynamic behavior of a one-dimensional thermoviscoelastic system

In this paper, we study the dynamic behavior of a one-dimensional linear thermoviscoelastic system with Dirichlet boundary conditions. A remarkable characteristic is that the system operator is not of compact resolvent. Using the asymptotic analysis technique, it is shown that there are three branches of eigenvalues: two of them are along the negative real axis approaching-∞ and another branch, distributing on the negative real axis, converges to a negative real point which is the unique continuous spectrum. Moreover, the set of generalized eigenfunctions forms a Riesz basis for the energy state space. Consequently, the spectrum-determined growth condition holds true, and an exponential stability is concluded. Finally, some numerical simulations are presented.