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

This paper aims to explore the long-term behavior of some nonlocal high-order-intime 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.