Singular Thermal Relaxation Limit for the Moore-Gibson-Thompson Equation Arising in Propagation of Acoustic Waves

TL;DR

This paper rigorously analyzes the asymptotic behavior of the Moore-Gibson-Thompson equation as the thermal relaxation coefficient approaches zero, showing convergence to the linearized Westervelt equation and exploring spectral properties.

Contribution

It provides a detailed asymptotic analysis and spectral study of the MGT equation in the limit of vanishing thermal relaxation, including convergence rates and stability results.

Findings

Strong convergence of solutions as { au} o 0

Identification of the limit dynamics as the linearized Westervelt equation

Spectral upper semicontinuity between MGT and Westervelt equations

Abstract

Moore-Gibson-Thompson (MGT) equations, which describe acoustic waves in a heterogeneous medium, are considered. These are the third order in time evolutions of a predominantly hyperbolic type. MGT models account for a finite speed propagation due to the appearance of thermal relaxation coefficient {\tau} {>} {0} in front of the third order time derivative. Since the values of {\tau} are relatively small and often negligible, it is important to understand the asymptotic behavior and characteristics of the model when {\tau} {\to} {0}. This is a particularly delicate issue since the {\tau}- dynamics is governed by a generator which is singular as {\tau} {\to} {0}. It turns out that the limit dynamics corresponds to the linearized Westervelt equation which is of a parabolic type. In this paper, we provide a rigorous analysis of the asymptotics which includes strong convergence of the corresponding evolutions over infinite horizon. This is obtained by studying convergence rates along with the uniform exponential stability of the third order evolutions. Spectral analysis for the MGT-equation along with a discussion of spectral uppersemicontinuity for both equations (MGT and linearized Westervelt) will also be provided.