Analysis of Solutions Regarding Vibrations of a Thermoelastic Layer

The initial boundary value problem for a system of partial differential equations is considered. The particular system of interest is the mathematical model describing vibrations of a loaded thermoelastic layer. The presence of a mixed derivative in the equations prevents the use of the standard method of variable separation. However, the Galerkin method may be used to construct solutions in the form of a functional series in terms of a system of trigonometric functions. It is established that the damping vibrations are exponential. The qualitative properties of the solutions are determined. An integral identity is derived and used to establish the uniqueness of the solution.