Energy decay of some multi-term nonlocal-in-time Moore--Gibson--Thompson equations

TL;DR

This paper investigates the long-term energy decay of nonlocal Moore--Gibson--Thompson equations, establishing conditions for exponential decay and analyzing the effects of memory kernels on system behavior.

Contribution

It provides new conditions ensuring exponential energy decay and refines well-posedness results for nonlocal high-order wave equations with memory effects.

Findings

Exponential decay of energy under certain kernel conditions

Energy vanishes with weaker kernel assumptions without a specific decay rate

Refined local well-posedness and initial-data compatibility conditions

Abstract

This paper aims to explore the long-term behavior of some nonlocal high-order-in-time wave equations. These equations, which have come to be known as Moore--Gibson--Thompson equations, arise in the context of acoustic wave propagation when taking into account thermal relaxation mechanisms in complex media such as human tissue. While the long-term behavior of linear local-in-time acoustic equations is well understood, their nonlocal counterparts still retain many mysteries. We establish here a set of assumptions that ensures exponential decay of the energy of the system. These assumptions are then shown to be verified by a large class of rapidly decaying memory kernels. Under weaker assumptions on the kernel we show that one may still obtain that the energy vanishes but without a rate of convergence. Furthermore, we refine previous results on the local well-posedness of the studied equation and establish a necessary initial-data compatibility condition for the solvability of the problem.