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 $Θ$ in a bounded interval $Ω\subset\mathbb{R}$. \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 $γ\equiv Γ$ 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 $γ,f\in C^2([0,\infty))$ as well as $Γ,F\in C^1([0,\infty))$ with $γ>0,Γ\ge0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of $γ$ and $f$, and further $|F(s)|\le C_F(1+s)^α$ for some $C_F>0$ and $α\in(0,1)$. In particular, for any given $T_\star$, initial mass $M$ and $0<\underlineγ<\overlineγ$, there exists a constant $δ_\star(M,T_\star,a,D, Ω, \underlineγ, \overlineγ,C_F,α)>0$, 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 $T_\star$.