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Based on low-temperature experimental data in solid dielectric crystals, we derive a model of heat conduction for rigid materials using the theory of thermo-dynamic internal state variables. The model is intended to admit wavelike propagation of heat below—and diffusive conduction above—a particular temperature value ϑ λ {\vartheta {_\lambda }} . A rapid decay of the speed of thermal waves occurs just below this temperature, coincident with the conductivity of the material reaching a peak. An analysis of weak and strong discontinuity waves is given in order to exhibit several main features of the proposed model.

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Abstract. Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, where dissipative effects are insignificant. Significantly, these features appear only at certain temperatures below which the materials reach their peak thermal conductivities (at approximately 18.5 K and 4.5 K for NaF and Bi, respectively). No wavelike behavior is found in NaF and Bi at higher temperatures, where only diffusive heat propagation is observed. Further, the speed, U E , at which small amplitude thermal waves propagate is a decreasing function of temperature in the region where the waves can be detected, after which the diffusion process dominates. This hyperbolic region appears separated from the diffusive region by a "critical" temperature, ϑ λ , at which U E = 0 [1]. The aim of this paper is to understand the dynamics of regular solutions having temperatures close to that of the phase transition. We begin, in section 2, by describing a phenomenological onedimensional model which uses an internal variable behaving as an order parameter. In section 3, we will examine properties of the phase transition, and in section 4, we obtain conditions under which this class of solutions remain smooth. Some explicit cases are, finally, examined in section 5.

Abstract. This paper discusses the finite time blow-up of the amplitude of acceleration waves in the case of heat propagation in one-dimensional rigid and elastic bodies. In both cases dissipation is not strong enough to preserve the smoothness of the solutions whose initial data is far from equilibrium.1. Introduction. Under suitable assumptions on the constitutive relations, the equations of isothermal nonlinear elasticity are of hyperbolic type. This fact can lead to the formation of shocks; that is, the velocity and deformation gradient become discontinuous, and for smooth data the Cauchy problem does not have a global smooth solution (Lax [9]). This strong effect of elastic nonlinearity is due to the fact that there is no dissipation. Many kinds of dissipation, e.g., Dafermos [5], will produce stabilizing phenomena. Such a stabilizing role is played, for example, by heat diffusion. This can be detected by investigating the evolution of the amplitude of acceleration waves along characteristics. Coleman and Gurtin [3,4] first showed, for inelastic materials with memory, how the amplitude of the waves can approach infinity in finite time.

This paper examines the effect of damping on a nonstrictly hyperbolic 2×2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

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