Large time stabilization of rough-data solutions in one-dimensional nonlinear thermoelasticity
Michael Winkler
TL;DR
The paper investigates the one-dimensional nonlinear thermoelastic system $u_{tt} = u_{xx} - (f(\Theta))_x$, $\Theta_t = \Theta_{xx} - f(\Theta) u_{xt}$ on a bounded interval with rough initial data, under energy- and entropy-based admissibility. It develops a strategy that leverages entropy dissipation and a Moser-type iteration to prove large-time stabilization of the temperature, yielding $\Theta(\cdot,t) \to \Theta_\infty$ in $L^{\infty}(\Omega)$ for some $\Theta_\infty>0$. Building on this, the authors show that the deformation variable vanishes in $L^{\infty}(\Omega)$ as $t\to\infty$ by analyzing $\omega$-limits and proving any limit must be trivial. The results establish robust asymptotic homogenization for rough data in a hyperbolic–parabolic thermoelastic system, governed by the energy and entropy principles, without requiring smoothing of initial data.
Abstract
In an open bounded real interval $Ω$, the model for one-dimensional thermoelasticity given by \[ u_{tt} = u_{xx} - \big(f(Θ)\big)_x, \qquad Θ_t = Θ_{xx} - f(Θ) u_{xt}, \] is considered along with homogeneous boundary conditions of Dirichlet type for $u$ and of Neumann type for $Θ$, under the assumption that $f\in C^1([0,\infty))$ satisfies $f(0)=0$, $f'\in L^\infty((0,\infty))$ and $f'>0$ on $[0,\infty)$. The focus is on initial data which are merely required to be consistent with the fundamental principles of energy conservation and entropy nondecrease, by satisfying \[ u_0\in W_0^{1,2}(Ω), u_{0t} \in L^2(Ω), 0 \le Θ_0\in L^1(Ω), Θ_0 \not\equiv 0. \] Despite an apparent lack of favorable compactness properties that have underlain previous related studies on more regular settings, it is shown that corresponding weak solutions stabilize in the sense that \[ \lim_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(Ω)}=0 \] and \[ {\rm ess} \lim_{\!\!\!\! t\to\infty} \|Θ(\cdot,t)-Θ_\infty\|_{L^\infty(Ω)}=0 \] with some $Θ_\infty>0$.