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

Authors

  • R. Marchand,

    1. Department of Mathematical Sciences, US Air Force Academy, Colorado Springs, CO, USA
    2. Department of Mathematics, Slippery Rock University, Slippery Rock, PA, USA
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  • T. McDevitt,

    1. Department of Mathematics and Computer Science, Elizabethtown College, PA, USA
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  • R. Triggiani

    Corresponding author
    1. Department of Mathematics and Statistics, KFUPM, Dhahran, Saudi Arabia
    • Department of Mathematics, University of Virginia, Charlottesville, VA, USA
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    • Research by R.T. was partially supported by the National Science Foundation under Grant DMS-0104305 and by the U.S. Air Force Office of Scientific Research under Grant FA 9550-09-1-0459.

R. Triggiani, Department of Mathematics, University of Virginia, Charlottesville, VA, USA.

E-mail: rt7u@virginia.edu

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.

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