Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters

Abstract

A Neumann-type initial-boundary value problem for

$$\begin{aligned} \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma (\Theta ) \nabla u_t) + a \nabla \cdot (\gamma (\Theta ) \nabla u) + \nabla \cdot f(\Theta ), \\ \Theta _t = D\Delta \Theta + \Gamma (\Theta ) |\nabla u_t|^2 + F(\Theta )\cdot \nabla u_t, \end{array} \right. \end{aligned}$$

is considered in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\), \(n\ge 1\). In the case when \(n=1\), \(\gamma \equiv \Gamma \) and \(f\equiv F\), this system coincides with the standard model for heat generation in a viscoelastic material of Kelvin-Voigt type, well-understood in situations in which \(\gamma =const\). Covering scenarios in which all key ingredients \(\gamma ,\Gamma ,f\) and F may depend on the temperature \(\Theta \) here, for initial data which merely satisfy \(u_0\in W^{1,p+2}(\Omega )\), \(u_{0t}\in W^{1,p}(\Omega )\) and \(\Theta _0\in W^{1,p}(\Omega )\) with some \(p\ge 2\) such that \(p>n\), a result on local-in-time existence and uniqueness is derived in a natural framework of weak solvability.