Large time existence in a thermoviscoelastic evolution problem with mildly temperature-dependent parameters

Abstract We consider $$\begin{aligned} \left\{ \begin{array}{l} u_{tt} = (\gamma (\Theta ) u_{xt})_x + a (\gamma (\Theta ) u_x)_x +(f(\Theta ))_x, \\ \Theta _t = D\Theta _{xx} + \Gamma (\Theta ) u_{xt}^2 + F(\Theta ) u_{xt}, \end{array}\right. \qquad \qquad (\star ) \end{aligned}$$ u tt = ( γ ( Θ ) u xt ) x + a ( γ ( Θ ) u x ) x + ( f ( Θ ) ) x , Θ t = D Θ xx + Γ ( Θ ) u xt 2 + F ( Θ ) u xt , ( ⋆ ) under Neumann boundary conditions for u and Dirichlet boundary conditions for $$\Theta $$ Θ in a bounded interval $$\Omega \subset \mathbb {R}$$ <mml: