The first real eigenvalue of a one-dimensional linear thermoelastic system

In this note, we show, for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions, that there is at least one real eigenvalue which is greater than the dominant eigenvalue of the “pure” heat equation with the same boundary conditions. The result concludes the spectrum-determined growth condition for the system by virtue of a result of Renardy [1]. Moreover, this property is shown to be preserved for the same system with boundary vibration control.