Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

In this paper we consider a two-dimensional hybrid thermo-elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the initial-boundary-value problem for the interactive system of partial differential equations which take account of the mechanical strains/stresses and the thermal stresses in the plate and the beam, can be associated with a uniformly bounded evolution operator. It turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system. This result yields solvability for the problem under optimal initial freedom of the displacement, velocity, and temperature in the plate and the beam, while uniform stability is readily available.