  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.
