American Institute of Mathematical Sciences

Advanced
September  2021, 10(3): 431-460. doi: 10.3934/eect.2020074

Decay rates for the Moore-Gibson-Thompson equation with memory

Hizia Bounadja 1, and  Belkacem Said Houari 2,,

1. 

Ferhat Abbas university, Setif, 19000, Algeria
2. 

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

* Corresponding author: Belkacem Said Houari, bhouari@sharjah.ac.ae

Received  August 2019 Revised  February 2020 Published  June 2020

The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces $ \mathbb{R}^{N} $. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [8] and [26], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the $ L^{2} $-norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.

Keywords: Moore–Gibson–Thompson equation, memory kernel, energy method, exponential decay, polynomial decay, regularity loss.
Mathematics Subject Classification: Primary: 35L80, 35B40, 74D05; Secondary: 49K20, 93D30, 93D20.
Citation: Hizia Bounadja, Belkacem Said Houari. Decay rates for the Moore-Gibson-Thompson equation with memory. Evolution Equations & Control Theory, 2021, 10 (3) : 431-460. doi: 10.3934/eect.2020074
References:
[1]

M. O. Alves, A. H. Caixeta, M. A. Jorge Silva and J. H. Rodrigues, Moore–Gibson–Thompson equation with memory in a history framework: a semigroup approach, Z. Angew. Math. Phys., 69 (2018), Paper No. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar

[2]

M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.  Google Scholar

[3]

J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. and Inf. Sciences., 9 (2015), 2233-2238.   Google Scholar

[4]

M. Conti, S. Gatti and V. Patta, Decay rates of Volterra equations on $\mathbb{R}^{N}$, Central European Journal of Mathematics, 5 (2007), 720-732.  doi: 10.2478/s11533-007-0024-2.  Google Scholar

[5]

F. Coulouvrat, On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.   Google Scholar

[6]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Guessmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar

[11]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete and Continuous Dynamical Systems B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[12]

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.   Google Scholar

[13]

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(1-34). doi: 10.1142/S0218202512500352.  Google Scholar

[14]

B. Kaltenbacher, Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar

[15]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar

[16]

I. Lasiecka, Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics, J. Evol. Equ., 17 (2017), 411-441.  doi: 10.1007/s00028-016-0353-3.  Google Scholar

[17]

I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8.  Google Scholar

[18]

I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅱ: General decay of energy, J. Differerential Equations., 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.  Google Scholar

[19]

W. Liu, Z. Chen and D. Chen, New general decay results for a Moore–Gibson–Thompson equation with memory, Applicable Analysis, 2019, 1–20. doi: 10.1080/00036811.2019.1577390.  Google Scholar

[20]

E. Mainini and G. Mola, Exponential and polynomial decay for first order linear volterra evolution equations, Postdoctoral Fellowship of the Japan Society for the promotion of Sciences, 67 (2009), 93-111.  doi: 10.1090/S0033-569X-09-01145-X.  Google Scholar

[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.  Google Scholar

[22]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters (ELSEVIER), 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.  Google Scholar

[23]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[24]

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.  Google Scholar

[25]

M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Eqs. and Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.  Google Scholar

[26]

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), 2017,447–478, https://arXiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8.  Google Scholar

[27]

M. Pellicer and B. Said-Houari, On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo, math. Appl., 2019, 1–39, https://arXiv.org/abs/1903.10181. Google Scholar

[28]

R. Racke and B. Said-Houari, Decay Rates for semilinear viscoelastic system in weighted spaces, Journal of Hyperbolic Differential Equations, 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.  Google Scholar

[29]

R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Konstanzer Schriften in Mathematik, 382 (2019), 127. Google Scholar

[30]

B. E. Treeby, J. Jiri, B. T. R. Alistair and P. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method, The Journal of the Acoustical Society of America, 131 (2012), 4324-4336.  doi: 10.1121/1.4712021.  Google Scholar

show all references

References:
[1]

M. O. Alves, A. H. Caixeta, M. A. Jorge Silva and J. H. Rodrigues, Moore–Gibson–Thompson equation with memory in a history framework: a semigroup approach, Z. Angew. Math. Phys., 69 (2018), Paper No. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar

[2]

M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.  Google Scholar

[3]

J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. and Inf. Sciences., 9 (2015), 2233-2238.   Google Scholar

[4]

M. Conti, S. Gatti and V. Patta, Decay rates of Volterra equations on $\mathbb{R}^{N}$, Central European Journal of Mathematics, 5 (2007), 720-732.  doi: 10.2478/s11533-007-0024-2.  Google Scholar

[5]

F. Coulouvrat, On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.   Google Scholar

[6]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Guessmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar

[11]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete and Continuous Dynamical Systems B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[12]

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.   Google Scholar

[13]

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(1-34). doi: 10.1142/S0218202512500352.  Google Scholar

[14]

B. Kaltenbacher, Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar

[15]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar

[16]

I. Lasiecka, Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics, J. Evol. Equ., 17 (2017), 411-441.  doi: 10.1007/s00028-016-0353-3.  Google Scholar

[17]

I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8.  Google Scholar

[18]

I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅱ: General decay of energy, J. Differerential Equations., 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.  Google Scholar

[19]

W. Liu, Z. Chen and D. Chen, New general decay results for a Moore–Gibson–Thompson equation with memory, Applicable Analysis, 2019, 1–20. doi: 10.1080/00036811.2019.1577390.  Google Scholar

[20]

E. Mainini and G. Mola, Exponential and polynomial decay for first order linear volterra evolution equations, Postdoctoral Fellowship of the Japan Society for the promotion of Sciences, 67 (2009), 93-111.  doi: 10.1090/S0033-569X-09-01145-X.  Google Scholar

[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.  Google Scholar

[22]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters (ELSEVIER), 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.  Google Scholar

[23]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[24]

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.  Google Scholar

[25]

M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Eqs. and Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.  Google Scholar

[26]

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), 2017,447–478, https://arXiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8.  Google Scholar

[27]

M. Pellicer and B. Said-Houari, On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo, math. Appl., 2019, 1–39, https://arXiv.org/abs/1903.10181. Google Scholar

[28]

R. Racke and B. Said-Houari, Decay Rates for semilinear viscoelastic system in weighted spaces, Journal of Hyperbolic Differential Equations, 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.  Google Scholar

[29]

R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Konstanzer Schriften in Mathematik, 382 (2019), 127. Google Scholar

[30]

B. E. Treeby, J. Jiri, B. T. R. Alistair and P. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method, The Journal of the Acoustical Society of America, 131 (2012), 4324-4336.  doi: 10.1121/1.4712021.  Google Scholar

[1]

Wenjun Liu, Zhijing Chen, Zhiyu Tu. New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory. Electronic Research Archive, 2020, 28 (1) : 433-457. doi: 10.3934/era.2020025
[2]

Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015
[3]

Arthur Henrique Caixeta, Irena Lasiecka, Valéria Neves Domingos Cavalcanti. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation. Evolution Equations & Control Theory, 2016, 5 (4) : 661-676. doi: 10.3934/eect.2016024
[4]

Wenhui Chen, Alessandro Palmieri. Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5513-5540. doi: 10.3934/dcds.2020236
[5]

Marta Pellicer, Joan Solà-Morales. Optimal scalar products in the Moore-Gibson-Thompson equation. Evolution Equations & Control Theory, 2019, 8 (1) : 203-220. doi: 10.3934/eect.2019011
[6]

Wenhui Chen, Alessandro Palmieri. A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020085
[7]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235
[8]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
[9]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017
[10]

Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503
[11]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559
[12]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197
[13]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773
[14]

Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic & Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531
[15]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[16]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[17]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001
[18]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[19]

Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445
[20]

Shikuan Mao, Yongqin Liu. Decay of solutions to generalized plate type equations with memory. Kinetic & Related Models, 2014, 7 (1) : 121-131. doi: 10.3934/krm.2014.7.121

2019 Impact Factor: 0.953

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences