Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy

We are interested in the Moore–Gibson–Thompson equation with memory τuttt+αutt+c2Au+bAut-∫0tg(t-s)Aw(s)ds=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau{u}_{ttt}+ \alpha u_{tt}+c^{2}\mathcal{A}u+b\mathcal{A}u_t -\int_0^{t}g(t-s)\mathcal{A} w(s){\rm {d}}s=0.$$\end{document}This model arises in high-frequency ultrasound applications accounting for thermal flux and molecular relaxation times. According to revisited extended irreversible thermodynamics, thermal flux relaxation leads to the third-order derivative in time while molecular relaxation leads to non-local effects governed by memory terms. The resulting model is of hyperbolic type with viscous effects. We first classify the memory into three types. Then, we study how a memory term creates damping mechanism and how the memory causes energy decay even in the cases when the original memoryless system is unstable.