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In acoustics, higher-order-in-time equations arise when taking into account aclass of thermal relaxation laws in the modeling of sound wave propagation. Inthis work, we analyze initial boundary value problems for a family of suchequations and determine the behavior of solutions as the relaxation timevanishes. In particular, we allow the leading term to be of fractional type.The studied model can be viewed as a generalization of the well-established(fractional) Moore--Gibson--Thompson equation with three, in general nonlocal,convolution terms involving two different kernels. The interplay of theseconvolutions will influence the uniform analysis and the limiting procedure. Tounify the theoretical treatment of this class of local and nonlocalhigher-order equations, we relax the classical assumption on the leading-termkernel and consider it to be a Radon measure. After establishing uniformwell-posedness with respect to the relaxation time of the considered generalmodel, we connect it, through a delicate singular limit procedure, tofractional second-order models of linear acoustics.

A unified analysis framework for generalized fractional Moore--Gibson--Thompson equations: Well-posedness and singular limits