Convexity considerations and spatial behavior for the harmonic vibrations in thermoelastic plates

In this paper we study the spatial behavior of the steady-state solutions for the approach of thin thermoelastic plates developed by Lagnese and Lions [J.E. Lagnese, J.-L. Lions, Modelling, Analysis and Control of Thin Plates, Collection RMA, vol. 6, Masson, Paris, 1988]. The model leads to a coupled complex system of partial differential equations, one of fourth order in terms of the amplitude of the vertical deflection and the other of second-order in terms of the amplitude of temperature field. Coupling in an appropriate way the two equations in an integral identity we are able to identify some cross-sectional line integral measures associated with the amplitudes of the vertical deflection and temperature vibrations, provided that the exciting frequency is less than a certain critical frequency. Furthermore, we are able to establish a second-order differential inequality whose integration furnishes a Saint-Venant type decay estimate for a bounded strip and an alternative of Phragmén-Lindelöf type for a semi-infinite strip. The critical frequency is individuated by means of the use of some Wirtinger and Knowles inequalities.