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

TL;DR

This paper establishes local existence and uniqueness of strong solutions for a quasilinear Moore-Gibson-Thompson type model describing thermoviscoelastic heat generation during acoustic wave propagation in a one-dimensional solid.

Contribution

It provides the first rigorous analysis of local strong solutions for this specific thermoviscoelastic evolution system with nonlinear coupling.

Findings

Proved local existence and uniqueness of solutions.

Identified conditions for regular initial data.

Established a framework for strong solvability in the model.

Abstract

This manuscript is concerned with the evolution system \[ \left\{ \begin{array}{l} u_{ttt} + \alpha u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + \big( \widehat{\gamma}(\Theta) u_x\big)_x, \Theta_t = D \Theta_{xx} + \Gamma(\Theta) u_{xt}^2, \end{array} \right. \] 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 $\gamma, \widehat{\gamma}$ and $\Gamma$ are sufficiently smooth with $\gamma>0, \widehat{\gamma}>0$ 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.