  In this paper, we study the classical thermoelastic system with Fourier's law
of heat conduction in the whole space $\mathbb{R}^n$ when $n=1,2,3$,
particularly, asymptotic profiles for its elastic displacement as large-time.
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,
under a new condition for weighted datum, 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 with weighted $L^1$ datum as a by-product. These results
imply that wave-structure large-time behaviors hold only for the one- and
two-dimensional thermoelastic systems.
