March  2020, 28(1): 433-457. doi: 10.3934/era.2020025

New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Wenjun Liu

Received  December 2019 Published  March 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant No. 11771216), the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725] and the Six Talent Peaks Project in Jiangsu Province (Grant No. 2015-XCL-020)

In this paper, we consider the fourth-order Moore-Gibson- Thompson equation with memory recently introduced by (Milan J. Math. 2017, 85: 215-234) that proposed the fourth-order model. We discuss the well-posedness of the solution by using Faedo-Galerkin method. On the other hand, for a class of relaxation functions satisfying $ g'(t)\leq-\xi(t)M(g(t)) $ for $ M $ to be increasing and convex function near the origin and $ \xi(t) $ to be a nonincreasing function, we establish the explicit and general energy decay result, from which we can improve the earlier related results.

Citation: 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
References:
[1]

M. O. Alves et al., Moore-Gibson-Thompson equation with memory in a history framework: A semigroup approach, Z. Angew. Math. Phys., 69 (2018), Art. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar

[2]

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A. H. CaixetaI. Lasiecka and V. N. D. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147.  doi: 10.1016/j.jde.2016.03.006.  Google Scholar

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A. H. CaixetaI. Lasiecka and V. N. Domingos Cavalcanti, On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evol. Equ. Control Theory, 5 (2016), 661-676.  doi: 10.3934/eect.2016024.  Google Scholar

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M. M. Guesmia and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations, 18 (2005), 583-600.   Google Scholar

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M. M. ChenW. J. Liu and W. C. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type Ⅲ with frictional damping and delay terms, Adv. Nonlinear Anal., 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.  Google Scholar

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F. Dell'OroI. 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

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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

[11]

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.  Google Scholar

[12]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb {R}^{n}$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.  Google Scholar

[13]

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

[14]

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.   Google Scholar

[15]

B. KaltenbacherI. 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.   Google Scholar

[16]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.  Google Scholar

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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.  Google Scholar

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G. LiX. Y. Kong and W. J. Liu, General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds, J. Integral Equations Appl., 30 (2018), 95-116.  doi: 10.1216/JIE-2018-30-1-95.  Google Scholar

[20]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

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W. J. Liu and Z. J. Chen, General decay rate for a Moore–Gibson–Thompson equation with infinite history, Z. Angew. Math. Phys., 71 (2020), Paper No. 43. doi: 10.1007/s00033-020-1265-1.  Google Scholar

[23]

W. J. Liu, Z. J. Chen and D. Q. Chen, New general decay results for a Moore-GibsonThompson equation with memory, Appl. Anal., in press, 2019. doi: 10.1080/00036811.2019.1577390.  Google Scholar

[24]

W. J. LiuK. W. Chen and J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform., 34 (2017), 521-542.  doi: 10.1093/imamci/dnv056.  Google Scholar

[25]

W. J. LiuD. H. Wang and D. Q. Chen, General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses, 41 (2018), 758-775.  doi: 10.1080/01495739.2018.1431826.  Google Scholar

[26]

W. J. Liu and W. F. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim., 80 (2019), 103-133.  doi: 10.1007/s00245-017-9460-y.  Google Scholar

[27]

A. MagañaA. Miranville and R. Quintanilla, On the time decay in phase lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.  Google Scholar

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S. A. Messaoudi and J. Hashim Hassan, General and optimal decay in a memory-type Timoshenko system, J. Integral Equations Appl., 30 (2018), 117-145.  doi: 10.1216/JIE-2018-30-1-117.  Google Scholar

[29]

S. A. Messaoudi and J. Hashim Hassan, New general decay results for a viscoelastic-type Timoshenko system, preprint. Google Scholar

[30]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects, Journal of the Aerospace Sciences, 27 (1960), 117-127.   Google Scholar

[31]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.  Google Scholar

[32]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.  Google Scholar

[33]

F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solution for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034.  Google Scholar

[34]

R. Z. Xu et al., Global well-posedness and global attractor of fourth order semilinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 1515-1529. doi: 10.1002/mma.3165.  Google Scholar

[35]

R. Z. Xu et al., The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649. doi: 10.3934/dcds.2017244.  Google Scholar

[36]

R. Z. Xu et al., Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[37]

R. Z. Xu et al., Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205. doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[38]

Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 (2010), 092703, 25 pp. doi: 10.1063/1.3477939.  Google Scholar

[39]

Z. J. Yang and B. X. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys. 50 (2009), no. 3, 032701, 29 pp. Google Scholar

show all references

References:
[1]

M. O. Alves et al., Moore-Gibson-Thompson equation with memory in a history framework: A semigroup approach, Z. Angew. Math. Phys., 69 (2018), Art. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

F. Boulanouar and S. Drabla, General boundary stabilization result of memory-type thermoelasticity with second sound, Electron. J. Differential Equations, 2014 (2014), 18 pp.  Google Scholar

[4]

A. H. CaixetaI. Lasiecka and V. N. D. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147.  doi: 10.1016/j.jde.2016.03.006.  Google Scholar

[5]

A. H. CaixetaI. Lasiecka and V. N. Domingos Cavalcanti, On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evol. Equ. Control Theory, 5 (2016), 661-676.  doi: 10.3934/eect.2016024.  Google Scholar

[6]

M. M. Guesmia and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations, 18 (2005), 583-600.   Google Scholar

[7]

M. M. ChenW. J. Liu and W. C. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type Ⅲ with frictional damping and delay terms, Adv. Nonlinear Anal., 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.  Google Scholar

[8]

J. A. ConejeroC. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. Inf. Sci., 9 (2015), 2233-2238.  doi: 10.12785/amis.  Google Scholar

[9]

F. Dell'OroI. 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

[10]

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

[11]

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.  Google Scholar

[12]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb {R}^{n}$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.  Google Scholar

[13]

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

[14]

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.   Google Scholar

[15]

B. KaltenbacherI. 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.   Google Scholar

[16]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.  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. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.  Google Scholar

[19]

G. LiX. Y. Kong and W. J. Liu, General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds, J. Integral Equations Appl., 30 (2018), 95-116.  doi: 10.1216/JIE-2018-30-1-95.  Google Scholar

[20]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969.  Google Scholar

[22]

W. J. Liu and Z. J. Chen, General decay rate for a Moore–Gibson–Thompson equation with infinite history, Z. Angew. Math. Phys., 71 (2020), Paper No. 43. doi: 10.1007/s00033-020-1265-1.  Google Scholar

[23]

W. J. Liu, Z. J. Chen and D. Q. Chen, New general decay results for a Moore-GibsonThompson equation with memory, Appl. Anal., in press, 2019. doi: 10.1080/00036811.2019.1577390.  Google Scholar

[24]

W. J. LiuK. W. Chen and J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform., 34 (2017), 521-542.  doi: 10.1093/imamci/dnv056.  Google Scholar

[25]

W. J. LiuD. H. Wang and D. Q. Chen, General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses, 41 (2018), 758-775.  doi: 10.1080/01495739.2018.1431826.  Google Scholar

[26]

W. J. Liu and W. F. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim., 80 (2019), 103-133.  doi: 10.1007/s00245-017-9460-y.  Google Scholar

[27]

A. MagañaA. Miranville and R. Quintanilla, On the time decay in phase lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.  Google Scholar

[28]

S. A. Messaoudi and J. Hashim Hassan, General and optimal decay in a memory-type Timoshenko system, J. Integral Equations Appl., 30 (2018), 117-145.  doi: 10.1216/JIE-2018-30-1-117.  Google Scholar

[29]

S. A. Messaoudi and J. Hashim Hassan, New general decay results for a viscoelastic-type Timoshenko system, preprint. Google Scholar

[30]

F. Moore and W. Gibson, Propagation of weak disturbances in a gas subject to relaxing effects, Journal of the Aerospace Sciences, 27 (1960), 117-127.   Google Scholar

[31]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.  Google Scholar

[32]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.  Google Scholar

[33]

F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solution for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034.  Google Scholar

[34]

R. Z. Xu et al., Global well-posedness and global attractor of fourth order semilinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 1515-1529. doi: 10.1002/mma.3165.  Google Scholar

[35]

R. Z. Xu et al., The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649. doi: 10.3934/dcds.2017244.  Google Scholar

[36]

R. Z. Xu et al., Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[37]

R. Z. Xu et al., Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205. doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[38]

Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 (2010), 092703, 25 pp. doi: 10.1063/1.3477939.  Google Scholar

[39]

Z. J. Yang and B. X. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys. 50 (2009), no. 3, 032701, 29 pp. Google Scholar

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