An abstract semigroup approach to the third‐order Moore–Gibson–Thompson partial differential equation arising in high‐intensity ultrasound: structural decomposition, spectral analysis, exponential stability

Abstract

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. Copyright © 2012 John Wiley & Sons, Ltd.