Classical Solutions to a Model for Heat Generation During Acoustic Wave Propagation in a Standard Linear Solid

Abstract

In an open bounded real interval, this manuscript studies the evolution system \begin{cases} u_{ttt}+\alpha u_{tt} = \big(\gamma(\Theta)u_{xt}\big)_x + \big(\hat{\gamma}(\Theta)u_x\big)_x, \\ \Theta_t = D\Theta_{xx} + \Gamma(\Theta)u_{xt}^2, \end{cases} which arises as a model for the generation of heat during propagation of acoustic waves in a standard linear solid.

A statement in local existence and uniqueness of classical solutions is derived for arbitrary $D>0$ and $\alpha\ge0$, for sufficiently smooth $\gamma,\hat{\gamma}$ and $\Gamma$ with $\gamma>0,\hat{\gamma}>0$ and $\Gamma\ge0$ on $[0,\infty)$, and for all suitably regular initial data of arbitrary size.