Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities

Abstract

An initial-boundary value problem for \[ \left\{ \begin{array}{ll} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \qquad & x\inΩ, \ t>0, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \qquad & x\inΩ, \ t>0, \end{array} \right. \] is considered in an open bounded real interval $Ω$. Under the assumption that $γ\in C^0([0,\infty))$ and $f\in C^0([0,\infty))$ are such that $f(0)=0$, and $k_γ\le γ\le K_γ$ as well as \[ |f(ξ)| \le K_f \cdot (ξ+1)^α \qquad \mbox{for all } ξ\ge 0 \] with some $k_γ>0, K_γ>0, K_f>0$ and $α<\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived.