Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity

TL;DR

This paper investigates a multidimensional thermoelasticity model, establishing existence, uniqueness, and long-term behavior of solutions, revealing temperature stabilization and displacement decomposition in higher dimensions.

Contribution

It extends a one-dimensional functional approach to higher dimensions, proving existence and asymptotic properties of solutions in multidimensional thermoelasticity.

Findings

Temperature stabilizes to a constant over time.

Displacement decomposes into oscillating divergence-free and converging curl-free parts.

Results apply to both two- and three-dimensional models.

Abstract

We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors (SIAM J.\ Math.\ Anal.\ \textbf{55} (2023), 7024--7038)) to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lam\'e operator are also stated.