Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

TL;DR

This paper analyzes the long-term behavior of solutions to the viscous Jordan-Moore-Gibson-Thompson equation in nonlinear acoustics, deriving asymptotic profiles and establishing global existence of small solutions with optimal estimates.

Contribution

It introduces refined asymptotic analysis for the JMGT equation and a new decomposition method for nonlinear terms, connecting the model to diffusion-waves.

Findings

Derived first- and second-order asymptotic profiles for solutions.

Established global existence of small Sobolev solutions.

Connected the JMGT equation behavior to diffusion-waves as time grows large.

Abstract

In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan-Moore-Gibson-Thompson (JMGT) equation in the whole space $\mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo's law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first- and second-order asymptotic profiles of solution to the Moore-Gibson-Thompson (MGT) equation as $t\gg 1$, which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as $t\gg1$.