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

Abstract

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