Local and Global Solvability in a Viscous Wave Equation Involving General Temperature-Dependence

Abstract

The present manuscript studies a coupled thermoelastic-viscoelastic system, modelling the interaction between mechanical displacement and temperature in viscoelastic materials in a bounded interval. This topic is of interest in the fields of applied mathematics and continuum mechanics. The system under consideration reads

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} u_{tt} = (\gamma (\Theta ) u_{xt})_{x} + (\tilde {\gamma }(\Theta ) u_{x})_{x}+(f( \Theta ))_{x}, \\ \Theta _{t} = D\Theta _{xx} + \Gamma (\Theta ) u_{xt}^{2}+F(\Theta )u_{xt}, \end{array}\displaystyle \right . \end{aligned}$$

in an open bounded interval, which with \(\gamma \equiv \tilde {\gamma }\equiv \Gamma \) as well as \(f\equiv F\) reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. In contrast to the preceding related studies, the present study focuses on situations in which not only \(f\) and \(F\), but also the core components \(\gamma \), \(\tilde {\gamma }\) and \(\Gamma \), are dependent on the temperature variable \(\Theta \). Firstly, a statement regarding the local existence of classical solutions is derived for arbitrary \(D > 0\), \(0 <\gamma \), \(\tilde {\gamma }\in C^{2}([0,\infty ))\), and \(0 \le \Gamma \in C^{1}([0, \infty ))\), for functions \(f\in C^{2}([0,\infty );\mathbb{R})\) and \(F\in C^{1}([0,\infty );\mathbb{R})\) with \(F(0)=0\), and for suitably regular initial data of arbitrary size. Secondly, if \(\tilde{\gamma } = a\cdot \gamma +\mu \), with \(a>0\) and arbitrary \(\mu >0\), there exists \(\delta >0\) with the property that whenever in addition to the above we have

$$ \frac{a}{\gamma (\Theta _{\star })}\le \delta \qquad \text{and}\qquad \frac{|f'(\Theta _{\star })|\cdot |F(\Theta _{\star })|}{D\cdot \gamma (\Theta _{\star })} \le \delta , $$

for initial data close to the constant level given by \(u = 0\) and \(\Theta =\Theta _{\star }\), with any fixed \(\Theta _{\star }\ge 0\), it is demonstrated that these solutions are indeed global in time and possess the property that \(u_{xt}\), \(u_{x}\), \(u_{xx}\) and \(\Theta _{x}\) decay exponentially fast in \(L^{2}\). In this context, the parameter \(\mu \) captures the weak inclusion of the electric field within the system. This aspect constitutes the primary novel contribution of the present analysis. The aforementioned results are obtained by detecting suitable dissipative properties of functionals involving norms of these gradients in \(L^{2}\) spaces.