Large-time asymptotic behavior for the classical thermoelastic system

Abstract

In this paper, we study the classical thermoelastic system with Fourier's law of heat conduction in the whole space Rn when n=1,2,3, and obtain asymptotic profiles of its elastic displacement when t is large. We discover optimal growth estimates of the elastic displacement when n=1,2, whose growth rates coincide with those for the free wave model, whereas when n=3 the optimal decay rate is related to the Gaussian kernel. Furthermore, the large-time optimal leading term is firstly introduced by the combination of diffusion-waves, the heat kernel and singular components. We also illustrate a second-order profile of solution by diffusion-waves as a by-product. These results imply that the thermoelastic system has the wave-structure for large t in the one- and two-dimensional cases only.