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Second sound thermoelastic stability of a string/beam structure

In this paper we study the one dimensional thermoelastic transmission problem in a special string/beam structure: the two components are coupled at an interface (identified to $0$). Either the string or the beam is supposed thermoelastic, the heat flux is given by the Cattaneo's law instead of the usual Fourier's law. We prove that the the energy decay of the whole system is exponential if the string is thermoelastic. When only the beam is thermoelastic, we prove that the energy of the coupling string/beam decays polynomially to zero as $\frac{1}{t}$ and the decay rate can be, at most, polynomially stable of order $\frac{1}{t^2}$.