Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities

The model $$\{\begin{array}{l}{u}_{tt}=(\gamma (\mathrm{\Theta })u_{xt}{)}_x+a{u}_{xx}-(f(\mathrm{\Theta }){)}_x,\\ {\mathrm{\Theta }}_t={\mathrm{\Theta }}_{xx}+\gamma (\mathrm{\Theta })u_{xt}^2-f(\mathrm{\Theta })u_{xt},\end{array}$$ for thermoviscoelastic evolution in one-dimensional Kelvin-Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if ${\gamma }_0>0$ is fixed, then there exists $\delta =\delta ({\gamma }_0)>0$ with the property that for suitably regular initial data of arbitrary size an associated initial-boundary value problem posed in an open bounded interval admits a global classical solution whenever $\gamma \in C^2([0,\mathrm{\infty }))$ and $f\in C^2([0,\mathrm{\infty }))$ are such that $f(0)=0$ and $\mathrm{\mid }f(\xi )\mathrm{\mid }\le K_f\cdot (\xi +1{)}^{\alpha }$ for all $\xi \ge 0$ and some $K_f>0$ and $\alpha <\frac{3}{2}$, and that $${\gamma }_0\le \gamma (\xi )\le {\gamma }_0+\delta forall\xi \ge 0.$$ This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.