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} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x +(f(\Theta))_x, \\[1mm] \Theta_t = D\Theta_{xx} + \Gamma(\Theta) u_{xt}^2 + F(\Theta) u_{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $\Theta$ in a bounded interval $\Omega\subset\mathbb{R}$. \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 $\gamma\equiv \Gamma$ 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 $\gamma>0,\Gamma\ge0$ 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 $$\underline\gamma \le\gamma\le \overline\gamma\quad\mbox{ and }\quad 0\le \Gamma\le \overline\gamma \quad \mbox{ as well as } \quad\|\gamma'\|_{L^\infty([0,\infty))}\le \delta_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le \delta_\star$$ hold, the maximal existence time of the classical solution to $(\star)$ surpasses $T_\star$.