Global smooth solutions in a one-dimensional thermoviscoelastic model with temperature-dependent paramaters

Abstract

This manuscript is concerned with the system \begin{align*} \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + (a(x,t) u_x)_x +(f(\Theta))_x, \\[1mm] \Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2 + f(\Theta) u_{xt}, \end{array} \right. \end{align*} which is used to describe thermoviscoelastic developments in one-dimensional Kelvin-Voigt materials. \abs It is assumed that $a,\gamma$ and $f$ are sufficiently smooth functions that satisfy $$c_\gamma<\gamma(\zeta) 0$ and $\alpha \in (0,5/6)$. Under these conditions, this study then establishes a result on the existence of global classical solutions for sufficiently smooth but arbitrarily large initial data.