  This manuscript considers the viscous/inviscid Moore-Gibson-Thompson (MGT)
equations with memory of type I in the whole space $\mathbb{R}^n$. For one
thing, associating with a new condition on initial data, we derive the optimal
$L^2$ estimates and the optimal leading term of the acoustic velocity potential
for large time, in which we analyze different contributions from viscous,
thermally relaxing, as well as hereditary fluids on large time asymptotic
behavior for the acoustic waves models. For another, via the multi-scale
analysis and energy methods in the Fourier space, we demonstrate the
$L^{\infty}$ inviscid limits in the sense of the diffusivity of sound tending
to zero, which match our WKB expansion of the solution. Finally, we give a
further application of our results on large time behavior for the nonlinear
Jordan-MGT equation in viscous hereditary fluids.
