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

We consider \begin{align*} \label{HS} \left{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})x + a (γ(Θ) u_x)x +(f(Θ))x, \[1mm] Θ_t = DΘ{xx} + Γ(Θ) u{xt}^2 + F(Θ) u{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $\mathrm{\Theta }$ in a bounded interval $\mathrm{\Omega }\subset \mathbb{R}$. \abs 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 $\gamma \equiv \mathrm{\Gamma }$ 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 $\gamma ,f\in C^2([0,\mathrm{\infty }))$ as well as $\mathrm{\Gamma },F\in C^1([0,\mathrm{\infty }))$ with $\gamma >0,\mathrm{\Gamma }\ge 0$ 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 $\mathrm{\mid }F(s)\mathrm{\mid }\le C_F(1+s{)}^{\alpha }$ for some $C_F>0$ and $\alpha \in (0,1)$. In particular, for any given $T_{\star }$, initial mass $M$ and $0<\underset{‾}{\gamma }<\bar{\gamma }$, there exists a constant ${\delta }_{\star }(M,T_{\star },a,D,\mathrm{\Omega },\underset{‾}{\gamma },\bar{\gamma },C_F,\alpha )>0$, such that if $$\underset{‾}{\gamma }\le \gamma \le \bar{\gamma }and0\le \mathrm{\Gamma }\le \bar{\gamma }aswellas\mathrm{\parallel }{\gamma }^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}([0,\mathrm{\infty }))}\le {\delta }_{\star }and\mathrm{\parallel }{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}([0,\mathrm{\infty }))}\le {\delta }_{\star }$$ hold, the maximal existence time of the classical solution to $(\star )$ surpasses $T_{\star }$.