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

TL;DR

This paper develops a unified analysis framework for generalized fractional Moore--Gibson--Thompson equations, establishing well-posedness and analyzing the singular limit as relaxation time approaches zero, encompassing local and nonlocal models.

Contribution

It introduces a generalized model with fractional and nonlocal terms, relaxing classical assumptions, and provides a unified approach to analyze well-posedness and singular limits.

Findings

Established uniform well-posedness with respect to relaxation time.

Connected the generalized model to fractional second-order acoustics models.

Handled both local and nonlocal convolution terms in the analysis.

Abstract

In acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In this work, we analyze initial boundary value problems for a family of such equations and determine the behavior of solutions as the relaxation time vanishes. 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 these convolutions will influence the uniform analysis and the limiting procedure. To unify the theoretical treatment of this class of local and nonlocal higher-order equations, we relax the classical assumption on the leading-term kernel and consider it to be a Radon measure. After establishing uniform well-posedness with respect to the relaxation time of the considered general model, we connect it, through a delicate singular limit procedure, to fractional second-order models of linear acoustics.