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This paper aims at exploring the long-term behavior of some nonlocalhigh-order-in-time wave equations. These equations, which have come to be knownas Moore--Gibson--Thompson equations, arise in the context of acoustic wavepropagation when taking into account thermal relaxation mechanisms in complexmedia such as human tissue. While the long-term behavior of linearlocal-in-time acoustic equations is well understood, their nonlocalcounterparts still retain many mysteries in that regard. We establish here aset of assumptions which ensures exponential decay of the energy of the system.These assumptions are then shown to be verified by a large class of rapidlydecaying memory kernels. Under weaker assumptions on the kernel we show thatone may still obtain that the energy vanishes but without a rate ofconvergence. We furthermore refine previous results on the local well-posednessof the studied equation and establish a necessary initial-data compatibilitycondition for the solvability of the problem.

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