Asymptotic analysis of a nonsimple thermoelastic rod

The asymptotic analysis of a one-dimensional nonsimple thermoe- lastic problem is considered in this paper. By a detailed spectral analysis, the asymptotic expressions for eigenvalues and eigenfunctions of the considered sys-Tem are developed. It is shown that the eigenfunctions form a Riesz basis on the Hilbert space and the eigenvalues asymptotically fall on two branches. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. This gives the spectrum-determined growth condition for the C₀-semigroup associated to the system, and consequently, the asymptotic and the exponential stability of the solutions are deduced. The approach developed in this paper confirms the already-existing results; furthermore, it can be extended to a larger field of applications such as coupled system of rod or beam with diffusion equation. The method will be illustrated by an example of thermoelastic beam equations with Dirichlet boundary conditions.