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August  2022, 15(8): 1941-1955. doi: 10.3934/dcdss.2022009

On the time decay for the MGT-type porosity problems

1. 

CINTECX, Departamento de Ingeniería Mecánica, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain

2. 

Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain

3. 

Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

*Corresponding author: Ramón Quintanilla

Received  October 2021 Revised  December 2021 Published  August 2022 Early access  January 2022

Fund Project: This paper is part of the projects PGC2018-096696-B-I00 and PID2019-105118GB-I00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER "A way to make Europe". The authors would also like to acknowledge the comments provided by the reviewer, which helped to improve the final quality of the paper

In this work we study three different dissipation mechanisms arising in the so-called Moore-Gibson-Thompson porosity. The three cases correspond to the MGT-porous hyperviscosity (fourth-order term), the MGT-porous viscosity (second-order term) and the MGT-porous weak viscosity (zeroth-order term). For all the cases, we prove that there exists a unique solution to the problem and we analyze the resulting point spectrum. We also show that there is an exponential energy decay for the first case, meanwhile for the second and third case only a polynomial decay is found. Finally, we present some one-dimensional numerical simulations to illustrate the behaviour of the discrete energy for each case.

Citation: Jacobo Baldonedo, José R. Fernández, Ramón Quintanilla. On the time decay for the MGT-type porosity problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1941-1955. doi: 10.3934/dcdss.2022009
References:
[1]

A. E. AbouelregalH. AhmadT. A. Nofal and H. Abu-Zinadah, Moore-Gibson-Thompson thermoelasticity model with temperature-dependent properties for thermo-viscoelastic orthotropic solid cylinder of infinite length under a temperature pulse, Phys. Scr., 96 (2021), 105201.  doi: 10.1088/1402-4896/abfd63.

[2]

J. Baldonedo, J. R. Fernández, A. Magaña and R. Quintanilla, Decay for strain gradient porous elastic waves, Submitted, (2021).

[3]

J. Baldonedo, J. R. Fernández and R. Quintanilla, Time decay for porosity problems, Math. Meth. Appl. Sci., early access, 2021. doi: 10.1002/mma.8054.

[4]

N. BazarraJ. R. FernándezA. Magaña and R. Quintanilla, A poro-thermoelastic problem with dissipative heat conduction, J. Thermal Stresses, 43 (2020), 1415-1436.  doi: 10.1080/01495739.2020.1780176.

[5]

N. Bazarra, J. R. Fernández and R. Quintanilla, Analysis of a Moore-Gibson-Thompson thermoelasticity problem, J. Comput. App. Math., 382 (2021), Paper No. 113058, 16 pp. doi: 10.1016/j.cam.2020.113058.

[6]

N. BazarraJ. R. Fernández and R. Quintanilla, On the decay of the energy for radial solutions in Moore-Gibson-Thompson thermoelasticity, Math. Mech. Solids, 26 (2021), 1507-1514.  doi: 10.1177/1081286521994258.

[7]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operators semigroups, Math. Annal., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[8]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems, In Handb. Numer. Anal., (eds. P. G. Ciarlet and J. L. Lions), 2 (1991), 17–351.

[10]

J. A. ConejeroC. Lizama and F. Ródenas, Chaotic behaviour of the solution of the Moore-Gibson-Thompson equation, Appl. Math. Inf. Sci., 9 (2015), 2233-2238. 

[11]

M. ContiV. PataM. Pellicer and R. Quintanilla, On the analiticity of the MGT-viscoelastic plate with heat conduction, J. Differential Equations, 269 (2020), 7862-7880.  doi: 10.1016/j.jde.2020.05.043.

[12]

M. ContiV. PataM. Pellicer and R. Quintanilla, A new approach to MGT-thermoviscoelasticity, Discrete Cont. Dyn. Sys., 41 (2021), 4645-4666.  doi: 10.3934/dcds.2021052.

[13]

M. ContiV. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptotic Anal., 120 (2020), 1-21.  doi: 10.3233/ASY-191576.

[14]

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.

[15]

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.

[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]

J. R. Fernández and R. Quintanilla, Moore-Gibson-Thompson theory for thermoelastic dielectrics, Appl. Math. Mech., 42 (2021), 309-316.  doi: 10.1007/s10483-021-2703-9.

[18]

D. Ieşan, A gradient theory of porous elastic solids, Z. Ang. Math. Mech., 100 (2020), e201900241, 18 pp. doi: 10.1002/zamm.201900241.

[19]

D. Ieşan and R. Quintanilla, On the theory of interacting continua with memory, J. Thermal Stresses, 25 (2002), 1161-1177.  doi: 10.1080/01495730290074586.

[20]

K. Jangid and S. Mukhopadhyay, A domain of influence theorem under MGT thermoelasticity theory, Math. Mech. Solids, 26 (2021), 285-295.  doi: 10.1177/1081286520946820.

[21]

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. 

[22]

H. Kumar and S. Mukhopadyay, Thermoelastic damping analysis in microbeam resonators based on Moore-Gibson-Thompson generalized thermoelasticity theory, Acta Mech., 231 (2020), 3003-3015.  doi: 10.1007/s00707-020-02688-6.

[23]

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.

[24]

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.

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999.

[26]

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.

[27]

M. Pellicer and R. Quintanilla, On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation, Zeit. Ang. Math. Phys., 71 (2020), Paper No. 84, 21 pp. doi: 10.1007/s00033-020-01307-7.

[28]

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.

[29]

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

[30]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[31]

R. Quintanilla, Moore-Gibson-Thompson thermoelasticity, Math. Mech. Solids, 24 (2019), 4020-4031.  doi: 10.1177/1081286519862007.

[32]

R. Quintanilla, Moore-Gibson-Thompson thermoelasticity with two temperatures, Appl. Engng. Sci., 1 (2020), 100006. 

[33]

P.A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.

show all references

References:
[1]

A. E. AbouelregalH. AhmadT. A. Nofal and H. Abu-Zinadah, Moore-Gibson-Thompson thermoelasticity model with temperature-dependent properties for thermo-viscoelastic orthotropic solid cylinder of infinite length under a temperature pulse, Phys. Scr., 96 (2021), 105201.  doi: 10.1088/1402-4896/abfd63.

[2]

J. Baldonedo, J. R. Fernández, A. Magaña and R. Quintanilla, Decay for strain gradient porous elastic waves, Submitted, (2021).

[3]

J. Baldonedo, J. R. Fernández and R. Quintanilla, Time decay for porosity problems, Math. Meth. Appl. Sci., early access, 2021. doi: 10.1002/mma.8054.

[4]

N. BazarraJ. R. FernándezA. Magaña and R. Quintanilla, A poro-thermoelastic problem with dissipative heat conduction, J. Thermal Stresses, 43 (2020), 1415-1436.  doi: 10.1080/01495739.2020.1780176.

[5]

N. Bazarra, J. R. Fernández and R. Quintanilla, Analysis of a Moore-Gibson-Thompson thermoelasticity problem, J. Comput. App. Math., 382 (2021), Paper No. 113058, 16 pp. doi: 10.1016/j.cam.2020.113058.

[6]

N. BazarraJ. R. Fernández and R. Quintanilla, On the decay of the energy for radial solutions in Moore-Gibson-Thompson thermoelasticity, Math. Mech. Solids, 26 (2021), 1507-1514.  doi: 10.1177/1081286521994258.

[7]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operators semigroups, Math. Annal., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[8]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems, In Handb. Numer. Anal., (eds. P. G. Ciarlet and J. L. Lions), 2 (1991), 17–351.

[10]

J. A. ConejeroC. Lizama and F. Ródenas, Chaotic behaviour of the solution of the Moore-Gibson-Thompson equation, Appl. Math. Inf. Sci., 9 (2015), 2233-2238. 

[11]

M. ContiV. PataM. Pellicer and R. Quintanilla, On the analiticity of the MGT-viscoelastic plate with heat conduction, J. Differential Equations, 269 (2020), 7862-7880.  doi: 10.1016/j.jde.2020.05.043.

[12]

M. ContiV. PataM. Pellicer and R. Quintanilla, A new approach to MGT-thermoviscoelasticity, Discrete Cont. Dyn. Sys., 41 (2021), 4645-4666.  doi: 10.3934/dcds.2021052.

[13]

M. ContiV. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptotic Anal., 120 (2020), 1-21.  doi: 10.3233/ASY-191576.

[14]

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.

[15]

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.

[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]

J. R. Fernández and R. Quintanilla, Moore-Gibson-Thompson theory for thermoelastic dielectrics, Appl. Math. Mech., 42 (2021), 309-316.  doi: 10.1007/s10483-021-2703-9.

[18]

D. Ieşan, A gradient theory of porous elastic solids, Z. Ang. Math. Mech., 100 (2020), e201900241, 18 pp. doi: 10.1002/zamm.201900241.

[19]

D. Ieşan and R. Quintanilla, On the theory of interacting continua with memory, J. Thermal Stresses, 25 (2002), 1161-1177.  doi: 10.1080/01495730290074586.

[20]

K. Jangid and S. Mukhopadhyay, A domain of influence theorem under MGT thermoelasticity theory, Math. Mech. Solids, 26 (2021), 285-295.  doi: 10.1177/1081286520946820.

[21]

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. 

[22]

H. Kumar and S. Mukhopadyay, Thermoelastic damping analysis in microbeam resonators based on Moore-Gibson-Thompson generalized thermoelasticity theory, Acta Mech., 231 (2020), 3003-3015.  doi: 10.1007/s00707-020-02688-6.

[23]

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.

[24]

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.

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999.

[26]

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.

[27]

M. Pellicer and R. Quintanilla, On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation, Zeit. Ang. Math. Phys., 71 (2020), Paper No. 84, 21 pp. doi: 10.1007/s00033-020-01307-7.

[28]

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.

[29]

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

[30]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[31]

R. Quintanilla, Moore-Gibson-Thompson thermoelasticity, Math. Mech. Solids, 24 (2019), 4020-4031.  doi: 10.1177/1081286519862007.

[32]

R. Quintanilla, Moore-Gibson-Thompson thermoelasticity with two temperatures, Appl. Engng. Sci., 1 (2020), 100006. 

[33]

P.A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.

Figure 1.  Roots behaviour for the fourth-order dissipation mechanism
Figure 2.  Roots behaviour for the second-order dissipation mechanism
Figure 3.  Roots behaviour for the zero-order dissipation mechanism
Figure 4.  Example 1: Dependence of the solution with respect to parameter $ \kappa $ (fourth-order dissipation mechanism- Case (i))
Figure 5.  Example 2: Dependence of the solution with respect to parameter $ a $ (second-order dissipation mechanism- Case (ii))
Figure 6.  Example 3: Dependence of the solution with respect to parameter $ \xi $ (zeroth-order dissipation mechanism- Case (iii))
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