# American Institute of Mathematical Sciences

June  2022, 15(6): 1355-1376. doi: 10.3934/dcdss.2022020

## Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

 1 Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstrasse 39, 10117 Berlin, Germany 2 Department of Mathematical Sciences, The University of Memphis, IBS, Polish Academy of Sciences, Warsaw

*Corresponding author: Marcelo Bongarti

Received  July 2021 Revised  November 2021 Published  June 2022 Early access  February 2022

Fund Project: The first author is supported by NSF grant DMS-1713506

The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).

Citation: Marcelo Bongarti, Irena Lasiecka, José H. Rodrigues. Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1355-1376. doi: 10.3934/dcdss.2022020
##### References:
 [1] M. Bongarti and I. Lasiecka, Boundary stabilization of the linear MGT equation with feedback Neumann control, In Deterministic and Stochastic Optimal Control and Inverse Problems, (eds. B. Jadamba, A. A. Khan, S. Migórski and M. Sama), CRC Press, (2021), 150–168. doi: 10.1201/9781003050575-7. [2] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGT equation from the boundary: Regularity and stabilization, Applicable Analysis, (2021), 1-39.  doi: 10.1080/00036811.2021.1999420. [3] F. Bucci and I. Lasiecka, Feedback control of the acoustic pressure in ultrasonic wave propagation, Optimization, 68 (2019), 1811-1854.  doi: 10.1080/02331934.2018.1504051. [4] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [5] C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, 247 (1958), 431. [6] C. Cattaneo, Sulla conduzione del calore, In Some Aspects of Diffusion Theory, (ed. A. Pignedoli), Springer Berlin Heidelberg, 42 (2011), 485–485. doi: 10.1007/978-3-642-11051-1_5. [7] T. Chen, T. Fan, W. Zhang, Y. Qiu, J. Tu, X. Guo and D. Zhang, Acoustic characterization of high intensity focused ultrasound fields generated from a transmitter with a large aperture, Journal of Applied Physics, 115 (2014), 114902.  doi: 10.1063/1.4868597. [8] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Physical Review Letters, 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301. [9] C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equations, J. Math. Anal. Appl., 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043. [10] J. A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation, Appl. Math. Inf. Sci., 9 (2015), 2233-2238. [11] F. Dell'Oro, I. Lasiecka and V. Pata, The Moore–Gibson–Thompson equation with memory in the critical case, J. Differential Equations, 261 (2016), 4188-4222.  doi: 10.1016/j.jde.2016.06.025. [12] F. Dell'Oro and V. Pata, On a fourth-order equation of Moore–Gibson–Thompson type, Milan J. Math., 85 (2017), 215-234.  doi: 10.1007/s00032-017-0270-0. [13] F. Ekoue, A. F. Halloy, D. Gigon, G. Plantamp and E. Zajdman, Maxwell-cattaneo regularization of heat equation, International Journal of Physical and Mathematical Sciences, 7 (2013), 772-776. [14] H. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Differential Equations, 5 (1969), 72-105.  doi: 10.1016/0022-0396(69)90105-3. [15] P. Jordan, Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons, The Journal of the Acoustical Society of America, 124 (2008), 2491-2491.  doi: 10.1121/1.4782790. [16] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189. [17] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [18] B. Kaltenbacher and I. Lasiecka, Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity, Palest. J. Math., 1 (2012), 1-10. [19] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. [20] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Wellposedness and exponential decay of the energy of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [21] B. Kaltenbacher and V. Nikolić, On the Jordan–Moore–Gibson–Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532. [22] I. Lasiecka and C. Lebiedzik, Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping, Control Cybernet., 28 (1999), 557-581. [23] I. Lasiecka and C. Lebiedzik, Asymptotic behaviour or nonlinear structural acoustic interactions with thermal effects on the interface, Nonlinear Anal., 49 (2002), 703-735.  doi: 10.1016/S0362-546X(01)00135-3. [24] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved neumann bc: Global uniqueness and observability in one shot, In Differential Geometric Methods in the Control of Partial Differential Equations, (eds. R. Gulliver, W. Littman and R. Triggiani), Providence, RI; American Mathematical Society; 1999, 268 (2000), 227–325. doi: 10.1090/conm/268/04315. [25] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part II: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052. [26] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. 1 of Die Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Heidelberg, 1972. [27] R. Marchand, T. McDevitt and R. Triggiani, 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, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1$^{st}$ edition, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011. [30] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [31] R. Spigler, More around Cattaneo equation to describe heat transfer processes, Math. Methods Appl. Sci., 43 (2020), 5953-5962.  doi: 10.1002/mma.6336. [32] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206.

show all references

##### References:
 [1] M. Bongarti and I. Lasiecka, Boundary stabilization of the linear MGT equation with feedback Neumann control, In Deterministic and Stochastic Optimal Control and Inverse Problems, (eds. B. Jadamba, A. A. Khan, S. Migórski and M. Sama), CRC Press, (2021), 150–168. doi: 10.1201/9781003050575-7. [2] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGT equation from the boundary: Regularity and stabilization, Applicable Analysis, (2021), 1-39.  doi: 10.1080/00036811.2021.1999420. [3] F. Bucci and I. Lasiecka, Feedback control of the acoustic pressure in ultrasonic wave propagation, Optimization, 68 (2019), 1811-1854.  doi: 10.1080/02331934.2018.1504051. [4] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [5] C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, 247 (1958), 431. [6] C. Cattaneo, Sulla conduzione del calore, In Some Aspects of Diffusion Theory, (ed. A. Pignedoli), Springer Berlin Heidelberg, 42 (2011), 485–485. doi: 10.1007/978-3-642-11051-1_5. [7] T. Chen, T. Fan, W. Zhang, Y. Qiu, J. Tu, X. Guo and D. Zhang, Acoustic characterization of high intensity focused ultrasound fields generated from a transmitter with a large aperture, Journal of Applied Physics, 115 (2014), 114902.  doi: 10.1063/1.4868597. [8] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Physical Review Letters, 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301. [9] C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equations, J. Math. Anal. Appl., 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043. [10] J. A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation, Appl. Math. Inf. Sci., 9 (2015), 2233-2238. [11] F. Dell'Oro, I. Lasiecka and V. Pata, The Moore–Gibson–Thompson equation with memory in the critical case, J. Differential Equations, 261 (2016), 4188-4222.  doi: 10.1016/j.jde.2016.06.025. [12] F. Dell'Oro and V. Pata, On a fourth-order equation of Moore–Gibson–Thompson type, Milan J. Math., 85 (2017), 215-234.  doi: 10.1007/s00032-017-0270-0. [13] F. Ekoue, A. F. Halloy, D. Gigon, G. Plantamp and E. Zajdman, Maxwell-cattaneo regularization of heat equation, International Journal of Physical and Mathematical Sciences, 7 (2013), 772-776. [14] H. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Differential Equations, 5 (1969), 72-105.  doi: 10.1016/0022-0396(69)90105-3. [15] P. Jordan, Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons, The Journal of the Acoustical Society of America, 124 (2008), 2491-2491.  doi: 10.1121/1.4782790. [16] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189. [17] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [18] B. Kaltenbacher and I. Lasiecka, Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity, Palest. J. Math., 1 (2012), 1-10. [19] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. [20] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Wellposedness and exponential decay of the energy of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [21] B. Kaltenbacher and V. Nikolić, On the Jordan–Moore–Gibson–Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532. [22] I. Lasiecka and C. Lebiedzik, Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping, Control Cybernet., 28 (1999), 557-581. [23] I. Lasiecka and C. Lebiedzik, Asymptotic behaviour or nonlinear structural acoustic interactions with thermal effects on the interface, Nonlinear Anal., 49 (2002), 703-735.  doi: 10.1016/S0362-546X(01)00135-3. [24] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved neumann bc: Global uniqueness and observability in one shot, In Differential Geometric Methods in the Control of Partial Differential Equations, (eds. R. Gulliver, W. Littman and R. Triggiani), Providence, RI; American Mathematical Society; 1999, 268 (2000), 227–325. doi: 10.1090/conm/268/04315. [25] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part II: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052. [26] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. 1 of Die Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Heidelberg, 1972. [27] R. Marchand, T. McDevitt and R. Triggiani, 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, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1$^{st}$ edition, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011. [30] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [31] R. Spigler, More around Cattaneo equation to describe heat transfer processes, Math. Methods Appl. Sci., 43 (2020), 5953-5962.  doi: 10.1002/mma.6336. [32] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206.
Illustration of the domain. The "red" convex portion of the boundary, in the context of HIFU, represents a device called transducer and its role is to concentrate the sound waves in the direction of the focus. The remaining part of the boundary represents an absorption area. (Font: B. Kaltenbacher)
Representation of the domain
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