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

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 $\Theta_\infty>0$.