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arXiv · 2309.07750

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

Mostafa Meliani·Belkacem Said-Houari
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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.

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Mostafa Meliani, Belkacem Said-Houari. 2024-08-08. Energy decay of some multi-term nonlocal-in-time Moore--Gibson--Thompson equations. https://arxiv.org/abs/2309.07750

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