The model \(\begin{aligned} \left\{ \begin{array}{l}u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + au_{xx} - \big (f(\Theta )\big )_x, \\[1mm] \Theta _t = \Theta _{xx} + \gamma (\Theta ) u_{xt}^2 - f(\Theta ) u_{xt}, \end{array} \right. \end{aligned}\) 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 \(\gamma \in C^2([0,\infty ))\) and \(f\in C^2([0,\infty ))\) are such that \(f(0)=0\) and \(|f(\xi )| \le K_f \cdot (\xi +1)^\alpha \) for all \(\xi \ge 0\) and some \(K_f>0\) and \(\alpha <\frac{3}{2}\) , and that \(\begin{aligned} \gamma _0 \le \gamma (\xi ) \le \gamma _0 + \delta \qquad \hbox {for all } \xi \ge 0. \end{aligned}\) 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.
