Large time asymptotic behavior for the weakly damped Jordan-Moore-Gibson-Thompson equation

TL;DR

This paper analyzes the large time behavior and decay properties of solutions to the weakly damped Jordan-Moore-Gibson-Thompson equation, revealing diffusion profiles and regularity-loss decay in different damping regimes.

Contribution

It provides the first comprehensive study of the asymptotic behavior of solutions to the weakly damped JMGT equation, highlighting differences from classical models.

Findings

Optimal decay estimates for solutions in all dimensions.

Diffusion profiles in the sub-critical damping case.

Regularity-loss decay in the critical damping case.

Abstract

This manuscript considers the Jordan-Moore-Gibson-Thompson (JMGT) equation and its linearized equation with an additional weak damping term (proposed by [B. Kaltenbacher, \emph{Inverse Problems} (2025)] firstly) in the whole space $\mathbb{R}^n$. We mainly study the unique existence and large time behavior, including optimal decay estimates and asymptotic profiles, of global in-time Sobolev solutions for any $n\geqslant 1$. This weak damping term leads to diffusion profiles in the sub-critical case $\delta>0$ and regularity-loss decay properties in the critical case $\delta=0$, which are greatly different from the results for the corresponding classical models without the weak damping term.