Summary: This paper considers an abstract third-order equation in a Hilbert space that is motivated by, and ultimately directed to, the “concrete” Moore-Gibson-Thompson equation arising in high-intensity ultrasound. In its simplest form, with certain specific values of the parameters, this third-order abstract equation (with unbounded free dynamical operator) is not well-posed. In general, however, in the present physical model, a suitable change of variable permits one to show that it has a special structural decomposition, with a precise, hyperbolic-dominated driving part. From this, various attractive dynamical properties follow: s.c. group generation; a refined spectral analysis to include a specifically identified point in the continuous spectrum of the generator (so that it does not have compact resolvent) as an accumulation point of eigenvalues; and a consequent theoretically precise exponential decay with the same decay rate in various function spaces. In particular, the latter is explicit and sharp up to a finite number of (stable) eigenvalues of finite multiplicity. A computer-based analysis confirms the theoretical spectral analysis findings. Moreover, it shows that the dynamic behavior of these unaccounted for finite-dimensional eigenvalues are the ones that ultimately may dictate the rate of exponential decay, and which can be estimated with arbitrarily preassigned accuracy.