We consider \(\begin{aligned} \left\{ \begin{array}{l} u_{tt} = (\gamma (\Theta ) u_{xt})_x + a (\gamma (\Theta ) u_x)_x +(f(\Theta ))_x, \\ \Theta _t = D\Theta _{xx} + \Gamma (\Theta ) u_{xt}^2 + F(\Theta ) u_{xt}, \end{array}\right. \qquad \qquad (\star ) \end{aligned}\) under Neumann boundary conditions for u and Dirichlet boundary conditions for \(\Theta \) in a bounded interval \(\Omega \subset \mathbb {R}\) . This model is a generalization of the classical system for the description of strain and temperature evolution in a thermo-viscoelastic material following a Kelvin-Voigt material law, in which \(\gamma \equiv \Gamma \) and \(f\equiv F\) . Different variations of this model have already been analyzed in the past and the present study draws upon a known result concerning the existence of classical solutions, which are local in time, for suitably smooth initial data, arbitrary \(a>0\) , \(D>0\) and \(\gamma ,f\in C^2([0,\infty ))\) as well as \(\Gamma ,F\in C^1([0,\infty ))\) with \(\gamma >0,\Gamma \ge 0\) and \(F(0)=0\) . Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of \(\gamma \) and f, and further \(|F(s)|\le C_F(1+s)^\alpha \) for some \(C_F>0\) and \(\alpha \in (0,1)\) . In particular, for any given \(T_\star \) , initial mass M and \(0< \underline{\gamma }<\overline{\gamma }\) , there exists a constant \(\delta _\star (M,T_\star ,a,D,\Omega ,\underline{\gamma },\overline{\gamma },C_F,\alpha )>0\) , such that if \(\underline{\gamma }\le \gamma \le \overline{\gamma }\quad \text{ and } \quad 0\le \Gamma \le \overline{\gamma }\) \( \text{ as } \text{ well } \text{ as } \quad \Vert \gamma '\Vert _{L^\infty ([0,\infty ))}\le \delta _\star \quad \text{ and } \quad \Vert f'\Vert _{L^\infty ([0,\infty ))}\le \delta _\star \) hold, the maximal existence time of the classical solution to \((\star )\) surpasses \(T_\star \) . Therefore, converting \((\star )\) into a parabolic system using the substitution \(v:=u_t+au\) is key to applying known methods from works on parabolic problems.
