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

Abstract

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 $\hat{I}˜$ in a bounded interval $\hat{I}\text{c◯}\subset \mathbb{R}$. \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 ${\hat{I}}^3\equiv \hat{I}“$ 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 ${\hat{I}}^3,f\in C^2([0,\infty ))$ as well as $\hat{I}“,F\in C^1([0,\infty ))$ with ${\hat{I}}^3>0,\hat{I}“\ge 0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of ${\hat{I}}^3$ and $f$, and further $|F(s)|\le C_F(1+s{)}^{\hat{I}}\pm$ for some $C_F>0$ and $\hat{I}\pm \in (0,1)$. In particular, for any given $T_{\star }$, initial mass $M$ and $0<{\underline{\hat{I}}}^3<{\overline{\hat{I}}}^3$, there exists a constant $\hat{I}{´}_{\star }(M,T_{\star },a,D,\hat{I}\text{c◯},{\underline{\hat{I}}}^3,{\overline{\hat{I}}}^3,C_F,\hat{I}\pm )>0$, such that if $$\underlineγ\leγ\le \overlineγ\quad\mbox{ and }\quad 0\le Γ\le \overlineγ\quad \mbox{ as well as } \quad\|γ'\|_{L^\infty([0,\infty))}\le δ_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le δ_\star $$ hold, the maximal existence time of the classical solution to $(\star )$ surpasses $T_{\star }$.