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

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 Rn. 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≫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≫1.