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

Abstract

In an open bounded real interval $\hat{I}\text{c◯}$, 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 $\hat{I}˜$, under the assumption that $f\in C^1([0,\infty ))$ satisfies $f(0)=0$, $f^{\prime }\in L^{\infty }((0,\infty ))$ and $f^{\prime }>0$ 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 $\hat{I}{˜}_{\infty }>0$.