Existence, uniqueness, and time-asymptotics of regular solutions in multidimensional thermoelasticity on domains with boundary

TL;DR

This paper establishes the existence, uniqueness, and long-term behavior of solutions to a nonlinear multidimensional thermoelasticity model with novel boundary conditions, revealing how displacement and temperature evolve over time.

Contribution

It introduces new boundary conditions for thermoelasticity, proves global solutions for small initial data, and analyzes the asymptotic behavior of displacement and temperature in multidimensional domains.

Findings

Existence and uniqueness of solutions for small initial data.

Temperature converges to a constant, and displacement's divergence-free part oscillates.

Temperature remains positive and tends to zero in the potential part.

Abstract

In the paper, we investigate the nonlinear thermoelasticity model in two- and three-dimensional convex and bounded domains. We propose new boundary conditions for the displacement. These conditions are not usual in thermoelasticity. Whereas, we posit the Neumann boundary condition for the temperature. We prove the existence of global, unique solutions for small initial data. The temperature positivity is also shown. Next, we investigate the long-time behavior of solutions. We show that the divergence-free part of the displacement oscillates. On the other hand, we prove that the potential part and the temperature are strongly coupled. The non-rotation part is heavily affected by heat propagation. It turns out that it tends to $0$ as $x$ approaches infinity. Additionally, the temperature converges to a constant function. Our techniques are firmly based on the functional $\F$ adopted from {\sc Bies, P. M., Cie\'slak, T., Fuest, M., Lankeit, J., Muha, B., and Trifunovi\'c, S.}, \emph{Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity}, arXiv: 2507.20794, 2025. The functional is based on the Fisher information and higher-order derivatives of the displacement. It responds well to the new boundary conditions. It allows us to close a priori estimates. We also need $L^{\infty}$-estimates for the temperature here. The Moser iterative procedure ensures it. The Helmholtz decomposition is applied to the displacement. The boundary conditions are crucial here. The boundary integrals that appear in the calculations at this point disappear thanks to these conditions. This allows us to split the problem into two separate ones. Each of them is associated with one part of the Helmholtz decomposition.