An Initial-Boundary Value Problem for Thermoelastic Plates

Let $$\bar{S} \times [ - {{h}_{0}}{\text{/2, }}{{h}_{0}}{\text{/2}}]$$ be a region in ℝ3occupied by a homogeneous and isotropic material, where S ⊂ ℝ2 is a finite domain bounded by a simple closed C2-curve of outward unit normal ν = (ν 1, ν 2)T and h 0 = const is the plate thickness. In the absence of body forces and moments, of forces and moments on the faces, and of body heat sources and sinks, the thermoelastic bending of the plate is described by the system [1], 10.1 $$B({{\partial }_{1}},{{\partial }_{2}},{{\partial }_{t}})u = 0,$$ where ∂α = ∂x α, α=1,2, ∂ t = ∂/∂t, u = (u 1, u 2, u 3, u 4)T, u 4 is the temperature, u 0 = (u 1, u 2, u 3)T characterizes the displacements [2], B is the matrix differential operator $$\left( {\begin{array}{*{20}{c}} { - \rho {{h}^{2}}\partial _{t}^{2} + {{A}_{{11}}}} & {{{A}_{{12}}}} & {{{A}_{{13}}}} & { - c{{\partial }_{1}}} \\ {{{A}_{{21}}}} & { - \rho {{h}^{2}}\partial _{t}^{2} + {{A}_{{22}}}} & {{{A}_{{23}}}} & { - c{{\partial }_{2}}} \\ {{{A}_{{31}}}} & {{{A}_{{32}}}} & { - \rho \partial _{t}^{2} + {{A}_{{33}}}} & 0 \\ { - {{h}^{2}}\eta {{\partial }_{t}}{{\partial }_{1}}} & { - {{h}^{2}}\eta {{\partial }_{t}}{{\partial }_{2}}} & 0 & {\mathcal{D} - {{s}^{{ - 1}}}{{\partial }_{t}}} \\ \end{array} } \right),$$ D is the Laplacian, A ij are the elements of the matrix A(∂ 1, ∂ 2) from the corresponding adiabatic equilibrium case [2], ρ is the density η and s are thermal coefficients, and h 1 = h 0 2 /12.