Thermoelasticity with antidissipation

Monica Conti; Lorenzo Liverani; Vittorino Pata

doi:10.3934/dcdss.2022040

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August 2022, 15(8): 2173-2188. doi: 10.3934/dcdss.2022040

Thermoelasticity with antidissipation

Monica Conti , Lorenzo Liverani and Vittorino Pata ^,

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

^*Corresponding author: Vittorino Pata

To Professor Maurizio Grasselli on his 60th birthday

Received December 2021 Published August 2022 Early access February 2022

We provide a complete stability analysis for the abstract differential system made by an antidamped wave-type equation, coupled with a dissipative heat-type equation

$ \begin{cases} u_{tt} + A u -\gamma u_t = p A^{\alpha} \theta \\ \theta_{t} + \kappa A^{\beta} \theta = - p A^{\alpha} u_t \end{cases} $

where

$ A $

is a strictly positive selfadjoint operator on a Hilbert space,

$ \gamma, \kappa>0 $

, and both the parameters

$ \alpha $

and

$ \beta $

can vary between

$ 0 $

and

$ 1 $

. The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping

$ \gamma $

and the damping

$ \kappa $

. Depending on the value of

$ (\alpha, \beta) $

in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as

$ t\to\infty $

Correction: Sections 7, 8 and 9 are missing from this article. Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template. They are added in Correction to “Thermoelasticity with antidissipation” (volume 15, number 8, 2022, 2173-2188).

Keywords: Thermoelasticity, Fourier heat conduction law, damping and antidamping, stability, exponential stability, exponential blow up.

Mathematics Subject Classification: Primary: 35B40, 35Q79; Secondary: 74B05, 74F05.

Citation: Monica Conti, Lorenzo Liverani, Vittorino Pata. Thermoelasticity with antidissipation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2173-2188. doi: 10.3934/dcdss.2022040

References:

[1]	F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749. doi: 10.1051/cocv/2016011.
[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]	F. Ammar-Khodja, A. Bader and A. Benabdallah, Dynamic stabilization of systems via decoupling techniques, ESAIM Control Optim. Calc. Var., 4 (1999), 577-593. doi: 10.1051/cocv:1999123.
[4]	K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007.
[5]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.
[6]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
[7]	G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977.
[8]	J. A. Burns, Z. Liu and S. M. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl., 179 (1993), 574-591. doi: 10.1006/jmaa.1993.1370.
[9]	M. Conti, L. Liverani and V. Pata, A note on the energy transfer in coupled differential systems, Commun. Pure Appl. Anal., 20 (2021), 1821-1831. doi: 10.3934/cpaa.2021042.
[10]	M. Conti, L. Liverani and V. Pata, The MGT-Fourier model in the supercritical case, J. Differential Equations, 301 (2021), 543-567. doi: 10.1016/j.jde.2021.08.030.
[11]	M. Conti, V. Pata, M. 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.
[12]	M. Conti, V. 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.
[13]	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.
[14]	J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159. doi: 10.1007/s00033-012-0274-0.
[15]	J. Hao, Z. Liu and J. Yong, Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations, J. Differential Equations, 259 (2015), 4763-4798. doi: 10.1016/j.jde.2015.06.010.
[16]	J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899. doi: 10.1137/0523047.
[17]	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.
[18]	Z.-Y. Liu and M. Renardy, A note on the equation of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6. doi: 10.1016/0893-9659(95)00020-Q.
[19]	Z. Liu and S. M. Zheng, Exponential energy decay of the Euler-Bernoulli beam with shear/thermal diffusion, J. Math. Anal. Appl., 196 (1995), 467-478. doi: 10.1006/jmaa.1995.1420.
[20]	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.
[21]	F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aerospace Sci., 27 (1960), 117-127. doi: 10.2514/8.8418.
[22]	J. E. Muñoz Rivera and R. Racke, Magneto-thermo-elasticity large-time behavior for linear systems, Adv. Differ. Equ., 6 (2001), 359-384.
[23]	J. E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Differential Equations, 127 (1996), 454-483. doi: 10.1006/jdeq.1996.0078.
[24]	V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513. doi: 10.1090/S0033-569X-06-01010-4.
[25]	V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3.
[26]	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.
[27]	W. Rudin, Functional Analysis, McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.
[28]	D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358. doi: 10.1006/jmaa.1993.1071.
[29]	P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.
[30]	R. Triggiani and J. Zhang, Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182. doi: 10.3934/eect.2018008.
[31]	J. Zabczyk, Mathematical Control Theory. An Introduction, Reprint of the 1995 edition, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4733-9.

show all references

References:

[1]	F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749. doi: 10.1051/cocv/2016011.
[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]	F. Ammar-Khodja, A. Bader and A. Benabdallah, Dynamic stabilization of systems via decoupling techniques, ESAIM Control Optim. Calc. Var., 4 (1999), 577-593. doi: 10.1051/cocv:1999123.
[4]	K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007.
[5]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.
[6]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
[7]	G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977.
[8]	J. A. Burns, Z. Liu and S. M. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl., 179 (1993), 574-591. doi: 10.1006/jmaa.1993.1370.
[9]	M. Conti, L. Liverani and V. Pata, A note on the energy transfer in coupled differential systems, Commun. Pure Appl. Anal., 20 (2021), 1821-1831. doi: 10.3934/cpaa.2021042.
[10]	M. Conti, L. Liverani and V. Pata, The MGT-Fourier model in the supercritical case, J. Differential Equations, 301 (2021), 543-567. doi: 10.1016/j.jde.2021.08.030.
[11]	M. Conti, V. Pata, M. 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.
[12]	M. Conti, V. 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.
[13]	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.
[14]	J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159. doi: 10.1007/s00033-012-0274-0.
[15]	J. Hao, Z. Liu and J. Yong, Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations, J. Differential Equations, 259 (2015), 4763-4798. doi: 10.1016/j.jde.2015.06.010.
[16]	J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899. doi: 10.1137/0523047.
[17]	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.
[18]	Z.-Y. Liu and M. Renardy, A note on the equation of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6. doi: 10.1016/0893-9659(95)00020-Q.
[19]	Z. Liu and S. M. Zheng, Exponential energy decay of the Euler-Bernoulli beam with shear/thermal diffusion, J. Math. Anal. Appl., 196 (1995), 467-478. doi: 10.1006/jmaa.1995.1420.
[20]	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.
[21]	F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aerospace Sci., 27 (1960), 117-127. doi: 10.2514/8.8418.
[22]	J. E. Muñoz Rivera and R. Racke, Magneto-thermo-elasticity large-time behavior for linear systems, Adv. Differ. Equ., 6 (2001), 359-384.
[23]	J. E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Differential Equations, 127 (1996), 454-483. doi: 10.1006/jdeq.1996.0078.
[24]	V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513. doi: 10.1090/S0033-569X-06-01010-4.
[25]	V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3.
[26]	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.
[27]	W. Rudin, Functional Analysis, McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.
[28]	D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358. doi: 10.1006/jmaa.1993.1071.
[29]	P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.
[30]	R. Triggiani and J. Zhang, Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182. doi: 10.3934/eect.2018008.
[31]	J. Zabczyk, Mathematical Control Theory. An Introduction, Reprint of the 1995 edition, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4733-9.

Figure 1. Regions of stability and instability of system (1.1)

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Figure 2. Regions $ \mathsf R_0' $ and $ \mathsf R_0'' $

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