Global smooth solutions in a one-dimensional thermoviscoelastic model with temperature-dependent paramaters

Abstract

This manuscript is concerned with the system \begin{align*} \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + (a(x,t) u_x)_x +(f(Θ))_x, \\[1mm] Θ_t = DΘ_{xx} + γ(Θ) u_{xt}^2 + f(Θ) u_{xt}, \end{array} \right. \end{align*} which is used to describe thermoviscoelastic developments in one-dimensional Kelvin-Voigt materials. \abs It is assumed that $a,{\hat{I}}^3$ and $f$ are sufficiently smooth functions that satisfy $$c_γ<γ(ζ)<C_γ, \quad γ''(ζ) \le 0,\quad f(0)=0, \quad |f'(ζ)|\le C_f \quad \mbox{ and } |f(ζ)|\le C_f(1+ζ)^α\quad \mbox{ for all }ζ\ge 0 $$ and some positive constants $c_{\hat{I}}^3,C_{\hat{I}}^3,C_f>0$ and $\hat{I}\pm \in (0,5/6)$. Under these conditions, this study then establishes a result on the existence of global classical solutions for sufficiently smooth but arbitrarily large initial data.