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Decay rates for the Moore-Gibson-Thompson equation with memory
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 |
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 [
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. |
[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. |
[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.
|
[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. |
[5] |
F. Coulouvrat,
On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.
|
[6] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[14] |
B. Kaltenbacher,
Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.
doi: 10.3934/eect.2015.4.447. |
[15] |
V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[23] |
V. Pata,
Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.
doi: 10.1007/s00032-009-0098-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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.
|
[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. |
[5] |
F. Coulouvrat,
On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.
|
[6] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[14] |
B. Kaltenbacher,
Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.
doi: 10.3934/eect.2015.4.447. |
[15] |
V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[23] |
V. Pata,
Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.
doi: 10.1007/s00032-009-0098-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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