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

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