Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities

An initial-boundary value problem for $$\{\begin{array}{ll}{u}_{tt}=(\gamma (\mathrm{\Theta })u_{xt}{)}_x+a{u}_{xx}-(f(\mathrm{\Theta }){)}_x,& x\in \mathrm{\Omega },t>0,\\ {\mathrm{\Theta }}_t={\mathrm{\Theta }}_{xx}+\gamma (\mathrm{\Theta })u_{xt}^2-f(\mathrm{\Theta })u_{xt},& x\in \mathrm{\Omega },t>0,\end{array}$$ is considered in an open bounded real interval $\mathrm{\Omega }$. Under the assumption that $\gamma \in C^0([0,\mathrm{\infty }))$ and $f\in C^0([0,\mathrm{\infty }))$ are such that $f(0)=0$, and $k_{\gamma }\le \gamma \le K_{\gamma }$ as well as $$\mathrm{\mid }f(\xi )\mathrm{\mid }\le K_f\cdot (\xi +1{)}^{\alpha }forall\xi \ge 0$$ with some $k_{\gamma }>0,K_{\gamma }>0,K_f>0$ and $\alpha <\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived.