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

In an open bounded real interval $\mathrm{\Omega }$, the model for one-dimensional thermoelasticity given by $$u_{tt}=u_{xx}-(f(\mathrm{\Theta }){)}_x,{\mathrm{\Theta }}_t={\mathrm{\Theta }}_{xx}-f(\mathrm{\Theta })u_{xt},$$ is considered along with homogeneous boundary conditions of Dirichlet type for $u$ and of Neumann type for $\mathrm{\Theta }$, under the assumption that $f\in C^1([0,\mathrm{\infty }))$ satisfies $f(0)=0$, $f^{\mathrm{\prime }}\in L^{\mathrm{\infty }}((0,\mathrm{\infty }))$ and $f^{\mathrm{\prime }}>0$ on $[0,\mathrm{\infty })$. 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 $$u_0\in W_0^{1,2}(\mathrm{\Omega }),u_{0t}\in L^2(\mathrm{\Omega }),0\le {\mathrm{\Theta }}_0\in L^1(\mathrm{\Omega }),{\mathrm{\Theta }}_0\equiv ̸0.$$ 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 $$\underset{t\to \mathrm{\infty }}{\lim }\mathrm{\parallel }u(\cdot ,t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}=0$$ and $$\mathrm{e}\mathrm{s}\mathrm{s}\underset{\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}t\to \mathrm{\infty }}{\lim }\mathrm{\parallel }\mathrm{\Theta }(\cdot ,t)-{\mathrm{\Theta }}_{\mathrm{\infty }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}=0$$ with some ${\mathrm{\Theta }}_{\mathrm{\infty }}>0$.