On the Moore–Gibson–Thompson Equation and Its Relation to Linear Viscoelasticity

Abstract

We discuss the parallel between the third-order Moore–Gibson–Thompson equation

$$\begin{aligned} {\partial _{ttt}} u + \alpha {\partial _{tt}}u-\beta \Delta {\partial _t} u - \gamma \Delta u =0 \end{aligned}$$

depending on the parameters $\alpha ,\beta ,\gamma >0,$ and the equation of linear viscoelasticity

$$\begin{aligned} \partial _{tt}u(t) - \kappa (0)\Delta u(t) - \int _{0}^\infty \kappa ^{\prime }(s)\Delta u(t-s)\,\mathrm{d}s=0 \end{aligned}$$

for the particular choice of the exponential kernel

$$\begin{aligned} \kappa (s) = a \mathrm{e}^{-b s} + c \end{aligned}$$

with $a,b,c>0$. In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel $\kappa$.