# American Institute of Mathematical Sciences

August  2022, 15(8): 1957-1985. doi: 10.3934/dcdss.2022107

## Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

 1 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10777, Berlin, Germany 2 Department of Mathematical Sciences, University of Memphis, 3725 Norriswood Ave, Memphis, TN 38152, United States, IBS, Polish Academy of Sciences, Warsaw

* Corresponding author: Irena Lasiecka

Received  December 2021 Revised  April 2022 Published  August 2022 Early access  May 2022

Fund Project: I. Lasiecka is supported by NSF grant DMS-1713506

Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.

Citation: Marcelo Bongarti, Irena Lasiecka. Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1957-1985. doi: 10.3934/dcdss.2022107
##### References:
 [1] P. R. Beesack, Gronwall Inequalities, Carleton University, Ottawa, Ont., 1975. [2] P. R. Beesack, On some Gronwall–type integral inequalities in n independent variables, J. Math. Anal. Appl., 100 (1984), 393-408.  doi: 10.1016/0022-247X(84)90089-1. [3] M. Bongarti, S. Charoenphon and I. Lasiecka, Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics, J. Evol. Equ., 21 (2021), 3553-3584.  doi: 10.1007/s00028-020-00654-2. [4] M. Bongarti and I. Lasiecka, Boundary stabilization of the linear MGT equation with feedback Neumann control, Deterministic and Stochastic Optimal Control and Inverse Problems, 7 (2021), 150-168. [5] M. Bongarti, I. Lasiecka and J. H. Rodrigues, Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity, Discrete and Continuous Dynamical Systems - S, 2022. doi: 10.3934/dcdss. 2022020. [6] M. Bongarti, I. Lasiecka and R. Triggianim, The SMGT equation from the boundary: Regularity and stabilization, Applicable Analysis, 101 (2022), 1735-1773.  doi: 10.1080/00036811.2021.1999420. [7] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, Comptes Rendus Mathématique, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [8] 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. [9] C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, 247 (1958), 431–433, https://ci.nii.ac.jp/naid/10018112216/en/. [10] C. Cattaneo, Sulla Conduzione Del Calore, In Aspects of Diffusion Theory, (2011), 485–485. [11] W. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, Evol. Equ. Control Theory, 10 (2021), 673-687.  doi: 10.3934/eect.2020085. [12] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301. [13] 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. [14] 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. [15] F. Dell'Oro, I. Lasiecka and V. Pata, A note on the Moore-Gibson-Thompson equation with memory of type Ⅱ, J. Evol. Equ., 20 (2020), 1251-1268.  doi: 10.1007/s00028-019-00554-0. [16] 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. [17] F. Dell'Oro and V. Pata, On the Moore–Gibson–Thompson Equation and its relation to linear viscoelasticity, Appl. Math. Optim., 76 (2017), 641-655.  doi: 10.1007/s00245-016-9365-1. [18] F. Ekoue, A. F. Halloy, D. Gigon, G. Plantamp and E. Zajdman, Maxwell-cattaneo regularization of heat equation, World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 7. [19] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [20] B. Kaltenbacher and C. Clyton, Avoiding degeneracy in the Westervelt equation by state constrained optimal control, Evol. Equ. Control Theory, 2 (2013), 281-300.  doi: 10.3934/eect.2013.2.281. [21] B. Kaltenbacher, C. Clayton and S. Veljović, Boundary optimal control of the westervalt and kuznetsov equations, JMAA, 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043. [22] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control and Cybernetics, 40 (2011), 971-988. [23] B. Kaltenbacher, I. Lasiecka and M. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear JMGT equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci, 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [24] 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. [25] B. Kaltenbacher and V. Nikolić, Vanishing relaxation time limit of the Jordan–Moore–Gibson–Thompson wave equation with Neumann and absorbing boundary conditions, Pure Appl. Funct. Anal., 5 (2020), 1-26. [26] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507–533, https://projecteuclid.org:443/euclid.die/1370378427. [27] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [28] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Volume 1, Cambridge University Press, Cambridge, 2000. [29] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann bc: Global uniqueness and observability in one shot, Contemp. Math., 268 (2000), 227-325.  doi: 10.1090/conm/268/04315. [30] 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. [31] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, ZAMP, 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8. [32] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications: Volume I, Springer-Verlag, Berlin, 1972. [33] 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. [34] V. Mazýa and T. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman (Advanced Publishing Program), Boston, MA, 1985. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] M. Pellicer and J. Solá-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011. [37] R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge-New York, 1982. [38] G. Savaré, Regularity and perturbation results for mixed second order elliptic equations, Comm. Partial Differential Equations, 22 (1997), 869-899.  doi: 10.1080/03605309708821287. [39] J. Simon, Compact sets in the space ${L}^p(0, {T}; {B})$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [40] R. Spigler, More around cattaneo equation to describe heat transfer processes, Math. Methods Appl. Sci., 43 (2020), 5953-5962.  doi: 10.1002/mma.6336. [41] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206. [42] D. Tataru, Boundary controllability of conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993. [43] R. Triggiani, Sharp interior and boundary regularity of the SMGTJ-equation with Dirichlet or Neumann boundary control, Springer Proc. Math. Stat., 325 (2020), 379-426.  doi: 10.1007/978-3-030-46079-2_22.

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##### References:
 [1] P. R. Beesack, Gronwall Inequalities, Carleton University, Ottawa, Ont., 1975. [2] P. R. Beesack, On some Gronwall–type integral inequalities in n independent variables, J. Math. Anal. Appl., 100 (1984), 393-408.  doi: 10.1016/0022-247X(84)90089-1. [3] M. Bongarti, S. Charoenphon and I. Lasiecka, Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics, J. Evol. Equ., 21 (2021), 3553-3584.  doi: 10.1007/s00028-020-00654-2. [4] M. Bongarti and I. Lasiecka, Boundary stabilization of the linear MGT equation with feedback Neumann control, Deterministic and Stochastic Optimal Control and Inverse Problems, 7 (2021), 150-168. [5] M. Bongarti, I. Lasiecka and J. H. Rodrigues, Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity, Discrete and Continuous Dynamical Systems - S, 2022. doi: 10.3934/dcdss. 2022020. [6] M. Bongarti, I. Lasiecka and R. Triggianim, The SMGT equation from the boundary: Regularity and stabilization, Applicable Analysis, 101 (2022), 1735-1773.  doi: 10.1080/00036811.2021.1999420. [7] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, Comptes Rendus Mathématique, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [8] 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. [9] C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, 247 (1958), 431–433, https://ci.nii.ac.jp/naid/10018112216/en/. [10] C. Cattaneo, Sulla Conduzione Del Calore, In Aspects of Diffusion Theory, (2011), 485–485. [11] W. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, Evol. Equ. Control Theory, 10 (2021), 673-687.  doi: 10.3934/eect.2020085. [12] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301. [13] 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. [14] 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. [15] F. Dell'Oro, I. Lasiecka and V. Pata, A note on the Moore-Gibson-Thompson equation with memory of type Ⅱ, J. Evol. Equ., 20 (2020), 1251-1268.  doi: 10.1007/s00028-019-00554-0. [16] 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. [17] F. Dell'Oro and V. Pata, On the Moore–Gibson–Thompson Equation and its relation to linear viscoelasticity, Appl. Math. Optim., 76 (2017), 641-655.  doi: 10.1007/s00245-016-9365-1. [18] F. Ekoue, A. F. Halloy, D. Gigon, G. Plantamp and E. Zajdman, Maxwell-cattaneo regularization of heat equation, World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 7. [19] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [20] B. Kaltenbacher and C. Clyton, Avoiding degeneracy in the Westervelt equation by state constrained optimal control, Evol. Equ. Control Theory, 2 (2013), 281-300.  doi: 10.3934/eect.2013.2.281. [21] B. Kaltenbacher, C. Clayton and S. Veljović, Boundary optimal control of the westervalt and kuznetsov equations, JMAA, 356 (2009), 738-751.  doi: 10.1016/j.jmaa.2009.03.043. [22] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control and Cybernetics, 40 (2011), 971-988. [23] B. Kaltenbacher, I. Lasiecka and M. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear JMGT equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci, 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [24] 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. [25] B. Kaltenbacher and V. Nikolić, Vanishing relaxation time limit of the Jordan–Moore–Gibson–Thompson wave equation with Neumann and absorbing boundary conditions, Pure Appl. Funct. Anal., 5 (2020), 1-26. [26] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507–533, https://projecteuclid.org:443/euclid.die/1370378427. [27] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224.  doi: 10.1007/BF01182480. [28] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Volume 1, Cambridge University Press, Cambridge, 2000. [29] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann bc: Global uniqueness and observability in one shot, Contemp. Math., 268 (2000), 227-325.  doi: 10.1090/conm/268/04315. [30] 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. [31] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, ZAMP, 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8. [32] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications: Volume I, Springer-Verlag, Berlin, 1972. [33] 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. [34] V. Mazýa and T. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman (Advanced Publishing Program), Boston, MA, 1985. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] M. Pellicer and J. Solá-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011. [37] R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge-New York, 1982. [38] G. Savaré, Regularity and perturbation results for mixed second order elliptic equations, Comm. Partial Differential Equations, 22 (1997), 869-899.  doi: 10.1080/03605309708821287. [39] J. Simon, Compact sets in the space ${L}^p(0, {T}; {B})$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [40] R. Spigler, More around cattaneo equation to describe heat transfer processes, Math. Methods Appl. Sci., 43 (2020), 5953-5962.  doi: 10.1002/mma.6336. [41] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206. [42] D. Tataru, Boundary controllability of conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993. [43] R. Triggiani, Sharp interior and boundary regularity of the SMGTJ-equation with Dirichlet or Neumann boundary control, Springer Proc. Math. Stat., 325 (2020), 379-426.  doi: 10.1007/978-3-030-46079-2_22.
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