  In this paper, we study the Cauchy problem for the linear and semilinear
Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the
linear MGT model, by utilizing WKB analysis associated with Fourier analysis,
we derive some $L^2$ estimates of solutions, which improve those in the
previous research [46]. Furthermore, asymptotic profiles of the solution and an
approximate relation in a framework of the weighted $L^1$ space are derived.
Next, with the aid of the classical energy method and Hardy's inequality, we
get singular limit results for an energy and the solution itself. Concerning
the semilinear MGT model, basing on these sharp $L^2$ estimates and
constructing time-weighted Sobolev spaces, we investigate global (in time)
existence of Sobolev solutions with different regularities. Finally, under a
sign assumption on initial data, nonexistence of global (in time) weak
solutions is proved by applying a test function method.
