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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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SUMMARYWe consider linear and non-linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave-like pulses travelling at ÿnite speed. This removal of the physical paradox of inÿnite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For di erent boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic-parabolic thermoelasticity is given. For Dirichlet type boundary conditions-rigidly clamped, constant temperature-the global existence of small, smooth solutions and the exponential stability are proved for a non-linear system.

Several thermomechanical models have been proposed from a heuristic point of view.\ud A mathematical analysis should help to clarify the applicability of these models, among\ud those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat\ud conduction models can be interpreted as formal expansions of delay equations. The delay\ud equations are shown to be ill-posed, as are the formal expansions of higher order in\ud contrast to lower-order expansions leading to Fourier's or Cattaneo's law. The ill-posedness\ud is proved, showing the lack of continuous dependence on the data, and thus showing that\ud these models (delay or higher-order expansion ones) are highly explosive. In this note we\ud shall present conditions for when this happens.Peer ReviewedPostprint (published version