Large time stabilization of rough-data solutions in one-dimensional nonlinear thermoelasticity

TL;DR

This paper proves that weak solutions to a one-dimensional thermoelasticity model with rough initial data stabilize over time, with displacement vanishing and temperature approaching a positive constant, despite limited regularity and compactness.

Contribution

It demonstrates long-term stabilization of weak solutions in a rough-data setting for a nonlinear thermoelasticity model, extending previous results to less regular initial conditions.

Findings

Displacement norm tends to zero as time approaches infinity.

Temperature converges uniformly to a positive steady state.

Stability holds despite limited initial regularity and compactness issues.

Abstract

In an open bounded real interval $\Omega$, the model for one-dimensional thermoelasticity given by \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \qquad \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered along with homogeneous boundary conditions of Dirichlet type for $u$ and of Neumann type for $\Theta$, 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 \[ u_0\in W_0^{1,2}(\Omega), u_{0t} \in L^2(\Omega), 0 \le \Theta_0\in L^1(\Omega), \Theta_0 \not\equiv 0. \] 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 \[ \lim_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=0 \] and \[ {\rm ess} \lim_{\!\!\!\! t\to\infty} \|\Theta(\cdot,t)-\Theta_\infty\|_{L^\infty(\Omega)}=0 \] with some $\Theta_\infty>0$.