Sharp large time asymptotic behavior for the multi-dimensional thermoelastic systems of type II and type III

In this paper, we study large time asymptotic behavior of the elastic displacement [Formula: see text] and the temperature difference [Formula: see text] for the thermoelastic systems of type II and type III in the whole space [Formula: see text] without using the thermal displacement transformation. For the type III model with hyperbolic thermal effect, we derive optimal growth/decay estimates and the novel double diffusion waves profiles for the solutions [Formula: see text] with the [Formula: see text] integrable initial data as large time, which improve the results in [X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Commun. Contemp. Math. 5(1) (2003) 25–83; M. Reissig and Y.-G. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable, Math. Methods Appl. Sci. 28(11) (2005) 1359–1381; L. Yang and Y.-G. Wang, Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D, Indiana Univ. Math. J. 55(4) (2006) 1333–1361; K. Jachmann, A unified treatment of models of thermoelasticity, Ph.D. thesis, TU Bergakademie Freiberg (2008)]. This hyperbolic thermal law produces a stronger singularity than the classical Fourier law in thermoelastic systems. For the type II model lacking of dissipation mechanism, we obtain optimal growth estimates and the new double waves profiles for the solutions [Formula: see text] as large time in low dimensions. Particularly, these results on the type II/III models show the infinite time blowup phenomena for [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] in the [Formula: see text] norm due to the hyperbolic influence. We also clarify competitions between the elastic waves effect and the hyperbolic thermal effect for the thermoelastic systems via the critical dimensions.