The first real eigenvalue of a one-dimensional linear thermoelastic system (Q1975713)

A one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions is considered. A rigorous proof of the conjecture that there is at least one real eigenvalue which is greater than the dominant eigenvalue of the ``pure'' heat conduction equation with the same boundary conditions. The validity of this result leads trivially to the spectrum-determined growth condition for this system. It is further proved that this property is also valid for the same thermoelastic system with boundary vibration control.