Asymptotic analysis of a nonsimple thermoelastic rod
Moncef Aouadi, Taoufik Moulahi · Discrete and Continuous Dynamical Systems - S · 2016
The asymptotic analysis of a one-dimensional nonsimple thermoelasticproblem is considered in this paper. By a detailed spectralanalysis, the asymptotic expressions for eigenvalues andeigenfunctions of the considered system are developed. It is shownthat the eigenfunctions form a Riesz basis on the Hilbert space andthe 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 parallelto the imaginary axis. This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, and consequently, the asymptotic and theexponential stability of the solutions are deduced. The approachdeveloped in this paper confirms the already-existing results; furthermore,it can be extended to a larger field of applications such ascoupled system of rod or beam with diffusion equation. The methodwill be illustrated by an example of thermoelastic beam equationswith Dirichlet boundary conditions.