2023/09/14 by Mostafa Meliani, Meliani, Mostafa, Belkacem Said‐Houari +1 · 1 citation
Engineering · Mathematics · #35A01 #35A02 #35B40 #35D30 #35L05 #35L35 #35R11 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2309.07750
openalex publication_date 2023/09/14 · openalex created_date 2023/09/16 · openalex updated_date 2026/07/28
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.