Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

Abstract

This manuscript is concerned with the evolution system \[ \left\{ \begin{array}{l} u_{ttt} + αu_{tt} = \big(γ(Θ) u_{xt}\big)_x + \big( \widehatγ(Θ) u_x\big)_x, Θ_t = D Θ_{xx} + Γ(Θ) u_{xt}^2, \end{array} \right. \] which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type. Under the assumptions that $D>0$ and $\alpha \ge 0$, and that $\gamma ,\hat{\gamma }$ and $\Gamma$ are sufficiently smooth with $\gamma >0,\hat{\gamma }>0$ and $\Gamma \ge 0$ on $[0,\infty )$, for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.