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 $u$ and the temperature difference $\theta$ for the thermoelastic systems of type II and type III in the whole space $\mathbb{R}^n$ 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 $u,\theta$ with the $L^1$ integrable initial data as large time, which improve the results in [32,26,31,17]. 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 $u,\theta$ as large time in low dimensions. Particularly, these results on the type II/III models show the infinite time blowup phenomena for $u$ if $n\leqslant 4$ and $\theta$ if $n\leqslant 2$ in the $L^2$ 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.