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

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.