We consider vibrating systems of hyperbolic Timoshenko type that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While proving exponential stability under the Fourier law of heat conduction, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier's law by changing to the Cattaneo law distroys the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property. 0 1 now regarding the heat flux vector as another function to be determined through the differential equation and initial and, in case, boundary conditions. The positive parameter τ is the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature. Combining (1.1) and (1.4) we obtain the hyperbolic, damped wave equation(1.5)Again, we obtain the well-known exponential stability. That is, both models, Fourier and Cattaneo, exhibit the same qualitative behavior, they both lead to exponentially stable systems for pure heat conduction. There are many coupled systems describing both the elastic behavior of a system as well as simultaneously the heat conduction within the system. Such thermoelastic systems have been treated by many authors, for a survey on classical thermoelasticity -classical here also indicating that the Fourier law for heat conduction is used -see e.g. [7]. It has been shown that spacially one-dimensional systems are, under appropriate boundary conditions or normalizations, exponentially stable in bounded reference configurations. In three space dimensions the same holds for radially symmetric situations.This has been extended to models where the Fourier law is replaced by the Cattaneo law in [14,15,10,5]. Moreover, it has been shown in the one-dimensional frame work, that, for real materials, the decay rates (type of the associated semigroup) of solutions to the both models are very close to each other, see [6], and that, again for real materials in the model of pulsed laser heating, differences for the displacement or the displacement gradient are of order 10 −5 m and 10 −10 m, respectively, cp. [5].These observations nourish the expectation that always both models lead to exponential stability (or both do not). We shall demonstrate for Timoshenko type systems that Fourier's law might predict exponential stability, while Cattaneo's law does not. This observation seems to be new and, maybe, unexpected. It turns out that for Timoshenko systems with history which are known to decay exponentially due to the history the introduction of a heat conduction via Fourier keeps this exponential decay property while the Cattaneo model even destroys this property.The first system we consider is the following coupling of two wave ...

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We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.

We consider a linear thermoelastic plate system where the heat flux is given by Cattaneo's model. We show that the resulting system is exponentially stable if and only if the rotational inertia coefficient μ is positive. On the other hand, for μ = 0, we establish polynomial stability results giving optimal rates of decay for initial data in the domain of the infinitesimal generator of the thermoelastic semigroup.

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