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

Abstract

The model \[ \left\{ \begin{array}{l} u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + au_{xx} - \big(f(\Theta)\big)_x, \\[1mm] \Theta_t = \Theta_{xx} + \gamma(\Theta) u_{xt}^2 - f(\Theta) u_{xt}, \end{array} \right. \] 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 \[ \gamma_0 \le \gamma(\xi) \le \gamma_0 + \delta \qquad \mbox{for all } \xi\ge 0. \] 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.