A note on asymptotic profiles for the thermoelastic plate system

TL;DR

This paper analyzes the long-term behavior of solutions to the thermoelastic plate system with Newton's cooling law, revealing how lower-order terms influence decay rates and asymptotic profiles in different spatial dimensions.

Contribution

It provides new insights into the asymptotic profiles and decay estimates for thermoelastic plates, especially highlighting the impact of lower-order temperature terms.

Findings

Optimal growth and decay estimates depending on dimension

Lower-order temperature terms weaken decay rates

Identification of new leading terms in asymptotic profiles

Abstract

We investigate the Cauchy problem for the thermoelastic plate system associated with Newton's law of cooling, where optimal growth ($n\leqslant 4$) or decay ($n\geqslant 5$) estimates and asymptotic profiles of solutions for large-time are studied. Especially, the additional lower-order term in the temperature equation weakens decay rates of the vertical displacement, and leads to a new leading term comparing with the classical thermoelastic plates.