  In this paper, we study asymptotic behaviors for classical thermoelastic
plate equations with the Fourier law of heat conduction in the whole space
$\mathbb{R}^n$, where we introduce a reduction methodology basing on
third-order (in time) differential equations and refined Fourier analysis. We
derive optimal growth estimates when $n\leqslant 3$, bounded estimates when
$n=4$, and decay estimates when $n\geqslant 5$ for the vertical displacement in
the $L^2$ norm. Particularly, the new critical dimension $n=4$ for
distinguishing the decisive role between the plate model and the Fourier law of
heat conduction is discovered. Moreover, concerning the small thermal parameter
in the temperature equation, we study the singular limit problem. We not only
show global (in time) convergence of the vertical displacements between
thermoelastic plates and structurally damped plates, but also rigorously
demonstrate a new second-order profile of the solution. Our methodology can
settle several closely related problems in thermoelasticity.
