Asymptotic behavior of the solution with positive temperature in nonlinear 3D thermoelasticity

TL;DR

This paper proves the global existence, uniqueness, and long-term convergence to equilibrium of solutions in a 3D nonlinear thermoelasticity system with positive temperature, using advanced energy and iteration techniques.

Contribution

It provides the first comprehensive proof of global well-posedness and asymptotic stability for the nonlinear 3D thermoelasticity system with positive temperature.

Findings

Solutions converge to equilibrium with uniform temperature

Temperature remains strictly positive over time

Established global existence and uniqueness of solutions

Abstract

In this paper, we study a hyperbolic-parabolic coupled system arising in nonlinear three-dimensional thermoelasticity. We establish the global well-posedness and asymptotic behavior of solutions. Our main result shows that, a thermoelastic body asymptotically converges to an equilibrium state with a uniform temperature distribution for every initial data, determined by energy conservation. The proof of the global well-posedness is divided into some steps. To begin with, we introduce an approximate problem and derive its solvability. Next, we establish a time-independent upper bound for the temperature via Moser iteration technique. Together with an estimate of gradient of entropy, we use a functional involving the Fisher information of the temperature, which enables us to handle a delicate Gronwall-type inequality, to obtain required estimates of the higher-order derivatives. Further, we prove the strict positivity of temperature by applying Moser iteration again on the negative part of the logarithm of the temperature, followed by a uniqueness argument for the weak solution. Finally, we define a dynamical system on a proper functional phase space and analyze the $\omega$-limit set for every initial data. This work provides a complete proof of the global well-posedness and the long-time behavior in the nonlinear three-dimensional thermoelasticity system.