# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022066
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## Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms

 Department of Mathematics, College of Sciences, University of Sharjah, P. O. Box: 27272, Sharjah, United Arab Emirates

Received  October 2021 Revised  April 2022 Early access May 2022

In this paper, we consider the 3D Jordan–Moore–Gibson–Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model removing the $L^1$-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

Citation: Belkacem Said-Houari. Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022066
##### References:
 [1] R. T. Beyer, Parameter of nonlinearity in fluids, The J. the Acoustical Society of America, 32 (1960), 719-721. [2] M. Bongarti, S. Charoenphon and I. Lasiecka, Singular thermal relaxation limit for the Moore–Gibson–Thompson equation arising in propagation of acoustic waves, Semigroups of Operators: Theory and Applications SOTA, 325 (2020), 147-182.  doi: 10.1007/978-3-030-46079-2_9. [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, 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, 15 (2022), 1355-1376.  doi: 10.3934/dcdss.2022020. [5] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGT equation from the boundary: Regularity and stabilization, Appl. Anal., 101 (2022), 1735-1773.  doi: 10.1080/00036811.2021.1999420. [6] 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. [7] 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. [8] F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore–Gibson–Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., 20 (2020), 837-867.  doi: 10.1007/s00028-019-00549-x. [9] W. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differential Equations, 292 (2021), 176-219.  doi: 10.1016/j.jde.2021.05.011. [10] F. A. Duck, Nonlinear acoustics in diagnostic ultrasound, Ultrasound in Medicine & Biology, 28 (2002), 1-18.  doi: 10.1016/S0301-5629(01)00463-X. [11] G. C. Gorain, Stabilization for the vibrations modeled by the 'standard linear model' of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8. [12] Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296. [13] M. F. Hamilton and D. T. Blackstock et al., Nonlinear Acoustics, Academic press San Diego, 1998. [14] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [15] 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. [16] B. Kaltenbacher and V. Nikolić, The inviscid limit of third-order linear and nonlinear acoustic equations, SIAM J. Appl. Math., 81 (2021), 1461-1482.  doi: 10.1137/21M139390X. [17] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay 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. [18] B. Kaltenbacher and V. Nikolić, 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. [19] V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics: Acoustics, 16 (1970), 467-470. [20] C. Lizama and S. Zamorano, Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics, J. Differential Equations, 266 (2019), 7813-7843.  doi: 10.1016/j.jde.2018.12.017. [21] 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. [22] J. Melchor, W. J. Parnell, N. Bochud, L. Peralta and G. Rus, Damage prediction via nonlinear ultrasound: A micro-mechanical approach, Ultrasonics, 93 (2019), 145-155.  doi: 10.1016/j.ultras.2018.10.009. [23] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [24] 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. [25] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8. [26] R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Commun. Contemp. Math., 23 (2021), Paper No. 2050069, 39 pp. doi: 10.1142/S0219199720500698. [27] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, American Mathematical Soc., 2006. doi: 10.1090/cbms/106. [28] J. Xu and S. Kawashima, The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218 (2015), 275-315.  doi: 10.1007/s00205-015-0860-3.

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##### References:
 [1] R. T. Beyer, Parameter of nonlinearity in fluids, The J. the Acoustical Society of America, 32 (1960), 719-721. [2] M. Bongarti, S. Charoenphon and I. Lasiecka, Singular thermal relaxation limit for the Moore–Gibson–Thompson equation arising in propagation of acoustic waves, Semigroups of Operators: Theory and Applications SOTA, 325 (2020), 147-182.  doi: 10.1007/978-3-030-46079-2_9. [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, 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, 15 (2022), 1355-1376.  doi: 10.3934/dcdss.2022020. [5] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGT equation from the boundary: Regularity and stabilization, Appl. Anal., 101 (2022), 1735-1773.  doi: 10.1080/00036811.2021.1999420. [6] 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. [7] 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. [8] F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore–Gibson–Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., 20 (2020), 837-867.  doi: 10.1007/s00028-019-00549-x. [9] W. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differential Equations, 292 (2021), 176-219.  doi: 10.1016/j.jde.2021.05.011. [10] F. A. Duck, Nonlinear acoustics in diagnostic ultrasound, Ultrasound in Medicine & Biology, 28 (2002), 1-18.  doi: 10.1016/S0301-5629(01)00463-X. [11] G. C. Gorain, Stabilization for the vibrations modeled by the 'standard linear model' of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8. [12] Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296. [13] M. F. Hamilton and D. T. Blackstock et al., Nonlinear Acoustics, Academic press San Diego, 1998. [14] B. Kaltenbacher, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447. [15] 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. [16] B. Kaltenbacher and V. Nikolić, The inviscid limit of third-order linear and nonlinear acoustic equations, SIAM J. Appl. Math., 81 (2021), 1461-1482.  doi: 10.1137/21M139390X. [17] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay 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. [18] B. Kaltenbacher and V. Nikolić, 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. [19] V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics: Acoustics, 16 (1970), 467-470. [20] C. Lizama and S. Zamorano, Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics, J. Differential Equations, 266 (2019), 7813-7843.  doi: 10.1016/j.jde.2018.12.017. [21] 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. [22] J. Melchor, W. J. Parnell, N. Bochud, L. Peralta and G. Rus, Damage prediction via nonlinear ultrasound: A micro-mechanical approach, Ultrasonics, 93 (2019), 145-155.  doi: 10.1016/j.ultras.2018.10.009. [23] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. [24] 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. [25] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8. [26] R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Commun. Contemp. Math., 23 (2021), Paper No. 2050069, 39 pp. doi: 10.1142/S0219199720500698. [27] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, American Mathematical Soc., 2006. doi: 10.1090/cbms/106. [28] J. Xu and S. Kawashima, The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218 (2015), 275-315.  doi: 10.1007/s00205-015-0860-3.
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