DECAY OF SOLUTIONS OF THE SYSTEM OF THERMOELASTICITY OF TYPE III

Abstract

This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space ℝn. For the first case, we introduce a decoupled system that allows to reduce the problem of the asymptotic behavior for the original system to a suitable observability inequality for the Lamé system. In this way most of the existing results for the classical system of thermoelasticity are shown to hold for this system too. In particular, we show that: (1) For most domains the energy of the system does not decay uniformly; (2) Under suitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two space dimensions, the energy of smooth solutions decays polynomially. For the problem in the whole space ℝn, first, based on Fourier analysis and Lyapunov's second method, we show that the energy of longitudinal and thermal waves of the system decays as that of the classical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decay rate of the high frequency longitudinal and thermal waves of the system.