Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the $L^q$ framework

TL;DR

This paper investigates the long-term behavior and asymptotic profiles of solutions to the viscous Moore-Gibson-Thompson equation using advanced analytical techniques, revealing decay estimates, singular limits, and initial layer formation.

Contribution

It introduces new $L^p-L^q$ decay estimates, asymptotic profiles, and a rigorous analysis of initial layer formation for the viscous MGT equation.

Findings

Derived $L^p-L^q$ decay estimates and asymptotic profiles.

Established global singular limits in the $L^q$ framework.

Justified the formation of initial layers under incompatible initial conditions.

Abstract

This manuscript studies some qualitative properties of solutions to the Cauchy problem for the viscous Moore-Gibson-Thompson (MGT) equation. For one thing, by applying the WKB analysis and diagonalization procedure, we derive some $L^p-L^q$ decay estimates and the large time asymptotic profile for a suitable energy term, which provides a new way to treat higher order MGT-type coupled systems. For another, we obtain the global (in time) singular limits in the $L^q$ framework and the higher order asymptotic profile with respect to small thermal relaxation via the multi-scale analysis and the Fourier analysis. Especially, provided the incompatible initial condition between the viscous MGT equation and the strongly damped wave equation, the formation of initial layer is rigorously justified.