Optimal large-time estimates and singular limits for thermoelastic plate equations with the Fourier law

TL;DR

This paper investigates the asymptotic behavior of thermoelastic plate equations with Fourier heat conduction, establishing optimal estimates, discovering a critical dimension, and analyzing singular limits with convergence and profile results.

Contribution

It introduces a reduction methodology based on third-order differential equations and Fourier analysis, providing new insights into the critical dimension and singular limits in thermoelasticity.

Findings

Optimal growth, bounded, and decay estimates depending on dimension

Discovery of a critical dimension n=4 for heat conduction effects

Global convergence and second-order profile in singular limit analysis

Abstract

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.