October  2021, 41(10): 4645-4666. doi: 10.3934/dcds.2021052

A new approach to MGT-thermoviscoelasticity

1. 

Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

2. 

Universitat de Girona - Departament d'Informàtica, Matemàtica Aplicada i Estadística, C. Maria Aurèlia Capmany, 41 (Campus Montilivi), 17003 Girona, Spain

3. 

Universitat Politècnica de Catalunya - Departament de Matemàtiques, C. Colom 11, 08222 Terrassa, Barcelona, Spain

* Corresponding author: Monica Conti

Received  September 2020 Revised  January 2021 Published  October 2021 Early access  April 2021

In this paper we discuss some thermoelastic and thermoviscoelastic models obtained from the Gurtin theory, based on the invariance of the entropy under time reversal. We derive differential systems where the temperature and the velocity are ruled by generalized versions of the Moore-Gibson-Thompson equation. In the one-dimensional case, we provide a complete analysis of the evolution, establishing an existence and uniqueness result valid for any choice of the constitutive parameters. This result turns out to be new also for the MGT equation itself. Then, under suitable assumptions on the parameters, corresponding to the subcritical regime of the system, we prove the exponential stability of the related semigroup.

Citation: Monica Conti, Vittorino Pata, Marta Pellicer, Ramon Quintanilla. A new approach to MGT-thermoviscoelasticity. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4645-4666. doi: 10.3934/dcds.2021052
References:
[1]

R. Borghesani and A. Morro, Relaxation functions and time-reversal invariance in thermal and electric conduction, Ann. Mat. Pura Appl., 114 (1977), 271-288.  doi: 10.1007/BF02413790.  Google Scholar

[2]

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.  Google Scholar

[3]

J. A. ConejeroC. Lizama and F. Ródenas, 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

[4]

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

[5]

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

[6]

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

[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 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

[9]

G. C. Gorain, Stabilization for the vibrations modeled by the standard linear model of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.), 4 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8.  Google Scholar

[10]

A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.   Google Scholar

[11]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.  doi: 10.1080/01495739208946136.  Google Scholar

[12]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. I. Classical continuum physics, Proc. Royal Society London A, 448 (1995), 335-356.  doi: 10.1098/rspa.1995.0020.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. II. Generalized continua, Proc. Royal Society London A, 448 (1995), 357-377.  doi: 10.1098/rspa.1995.0021.  Google Scholar

[15]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. III. Mixtures of interacting continua, Proc. Royal Society London A, 448 (1995), 379-388.  doi: 10.1098/rspa.1995.0022.  Google Scholar

[16]

M. E. Gurtin, Time-reversal and symmetry in the thermodynamics of materials with memory, Arch. Rational Mech. Anal., 44 (1971/72), 387-399.  doi: 10.1007/BF00249968.  Google Scholar

[17]

D. Ieȿan, On the theory of thermoelasticity without energy dissipation, J. Thermal Stresses, 21 (1998), 295-307.  doi: 10.1080/01495739808956148.  Google Scholar

[18]

D. Ieȿan, Thermopiezoelectricity without energy dissipation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 631-656.  doi: 10.1098/rspa.2007.0264.  Google Scholar

[19]

D. Ieȿan and A. Scalia, Some theorems in the theory of thermoviscoelasticity, J. Thermal Stresses, 12 (1989), 225-239.  doi: 10.1080/01495738908961963.  Google Scholar

[20]

R. J. Knops and E. W. Wilkes, Theory of Elastic Stability, Encyclopedia of Physics VIa/3, Springer-Verlag, Berlin, 1973,125-302.  Google Scholar

[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.   Google Scholar

[22]

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

[23]

M. C. LeseduarteA. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375-391.  doi: 10.3934/dcdsb.2010.13.375.  Google Scholar

[24]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299-309.   Google Scholar

[25]

A. Magaña and R. Quintanilla, On the existence and uniqueness in phase-lag thermoelasticity, Meccanica, 53 (2018), 125-134.  doi: 10.1007/s11012-017-0727-9.  Google Scholar

[26]

R. MarchandT. 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

[27]

A. Miranville and R. Quintanilla, Exponential decay in one-dimensional type II thermoviscoelasticity with voids, J. Comput. Appl. Math., 368 (2020), n.112573. doi: 10.1016/j.cam.2019.112573.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

M. Pellicer and R. Quintanilla, On uniqueness and instability for some thermoemechanical problems involving the Moore-Gibson-Thompson equation, Z. Angew. Math. Phys, 71 (2020). doi: 10.1007/s00033-020-01307-7.  Google Scholar

[30]

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), 447-479.  doi: 10.1007/s00245-017-9471-8.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Prüss, On the spectrum of ${C}_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar

[33]

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

[34]

A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York, 1958.  Google Scholar

[35]

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

show all references

References:
[1]

R. Borghesani and A. Morro, Relaxation functions and time-reversal invariance in thermal and electric conduction, Ann. Mat. Pura Appl., 114 (1977), 271-288.  doi: 10.1007/BF02413790.  Google Scholar

[2]

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.  Google Scholar

[3]

J. A. ConejeroC. Lizama and F. Ródenas, 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

[4]

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

[5]

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

[6]

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

[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 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

[9]

G. C. Gorain, Stabilization for the vibrations modeled by the standard linear model of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.), 4 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8.  Google Scholar

[10]

A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.   Google Scholar

[11]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264.  doi: 10.1080/01495739208946136.  Google Scholar

[12]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. I. Classical continuum physics, Proc. Royal Society London A, 448 (1995), 335-356.  doi: 10.1098/rspa.1995.0020.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. II. Generalized continua, Proc. Royal Society London A, 448 (1995), 357-377.  doi: 10.1098/rspa.1995.0021.  Google Scholar

[15]

A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media. III. Mixtures of interacting continua, Proc. Royal Society London A, 448 (1995), 379-388.  doi: 10.1098/rspa.1995.0022.  Google Scholar

[16]

M. E. Gurtin, Time-reversal and symmetry in the thermodynamics of materials with memory, Arch. Rational Mech. Anal., 44 (1971/72), 387-399.  doi: 10.1007/BF00249968.  Google Scholar

[17]

D. Ieȿan, On the theory of thermoelasticity without energy dissipation, J. Thermal Stresses, 21 (1998), 295-307.  doi: 10.1080/01495739808956148.  Google Scholar

[18]

D. Ieȿan, Thermopiezoelectricity without energy dissipation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 631-656.  doi: 10.1098/rspa.2007.0264.  Google Scholar

[19]

D. Ieȿan and A. Scalia, Some theorems in the theory of thermoviscoelasticity, J. Thermal Stresses, 12 (1989), 225-239.  doi: 10.1080/01495738908961963.  Google Scholar

[20]

R. J. Knops and E. W. Wilkes, Theory of Elastic Stability, Encyclopedia of Physics VIa/3, Springer-Verlag, Berlin, 1973,125-302.  Google Scholar

[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.   Google Scholar

[22]

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

[23]

M. C. LeseduarteA. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375-391.  doi: 10.3934/dcdsb.2010.13.375.  Google Scholar

[24]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299-309.   Google Scholar

[25]

A. Magaña and R. Quintanilla, On the existence and uniqueness in phase-lag thermoelasticity, Meccanica, 53 (2018), 125-134.  doi: 10.1007/s11012-017-0727-9.  Google Scholar

[26]

R. MarchandT. 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

[27]

A. Miranville and R. Quintanilla, Exponential decay in one-dimensional type II thermoviscoelasticity with voids, J. Comput. Appl. Math., 368 (2020), n.112573. doi: 10.1016/j.cam.2019.112573.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

M. Pellicer and R. Quintanilla, On uniqueness and instability for some thermoemechanical problems involving the Moore-Gibson-Thompson equation, Z. Angew. Math. Phys, 71 (2020). doi: 10.1007/s00033-020-01307-7.  Google Scholar

[30]

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), 447-479.  doi: 10.1007/s00245-017-9471-8.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Prüss, On the spectrum of ${C}_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar

[33]

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

[34]

A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York, 1958.  Google Scholar

[35]

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

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