December  2017, 37(12): 6333-6352. doi: 10.3934/dcds.2017274

Stability for thermoelastic plates with two temperatures

1. 

Department of Mathematics, UPC Terrassa, Colom 11, 08222 Terrassa, Spain

2. 

Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

* Corresponding author

Received  January 2017 Revised  June 2017 Published  August 2017

Fund Project: The first author is supported by the project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P) (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -with Fourier or with Cattaneo type heat conduction -we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability.

Citation: Ramón Quintanilla, Reinhard Racke. Stability for thermoelastic plates with two temperatures. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6333-6352. doi: 10.3934/dcds.2017274
References:
[1]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Instit. Mat. Univ. Trieste Suppl., 28 (1996), 1-28. 

[2]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. 

[3]

P. J. ChenM. E. Gurtin and W. O. Williams, A note on non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970.  doi: 10.1007/BF01602278.

[4]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112. 

[5]

R. Denk and R. Racke, $L^p$ resolvent estimates and time decay for generalized thermoelastic plate equations, Electronic J. Differential Equations, 48 (2006), 1-16. 

[6]

R. DenkR. Racke and Y. Shibata, $L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains, Advances Differential Equations, 14 (2009), 685-715. 

[7]

R. DenkR. Racke and Y. Shibata, Local energy decay estimate of solutions to the thermoelastic plate equations in two-and three-dimensional exterior domains, J. Analysis Appl., 29 (2010), 21-62.  doi: 10.4171/ZAA/1396.

[8]

H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound, J. Math. Phys. , 53 (2012), 073509, 13 pp. doi: 10.1063/1.4734239.

[9]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems --Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.

[10]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394.  doi: 10.1090/S0002-9947-1978-0461206-1.

[11]

F. Huang, Characteristic conditions for exponential staility of linear dynamical systems in Hilbert spaces, Ann. Diff. Equations, 1 (1985), 43-56. 

[12]

B. Kabil, Zur Asymptotik bei Resonator-Gleichungen, Diplomarbeit (diploma thesis), University of Konstanz, 2011.

[13]

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.

[14]

I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermoelastic equations, Advances Differential Equations, 3 (1998), 387-416. 

[15]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM, Proc., 4 (1998), 199-222.  doi: 10.1051/proc:1998029.

[16]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions, Abstract Appl. Anal., 3 (1998), 153-169.  doi: 10.1155/S1085337598000487.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457-482. 

[18]

M. C. LeseduarteR. Quintanilla and R. Racke, On non-exponential decay in generalized thermoelasticity with two temperatures, Appl. Math. Letters, 70 (2017), 18-25.  doi: 10.1016/j.aml.2017.02.020.

[19]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.

[20]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Res. Notes Math., 1999.

[21]

J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058.

[22]

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.

[23]

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

[24]

R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures, Dynamics Continous, Discrete Impulsive Sys., Ser. A: Math. Anal., 11 (2004), 57-68. 

[25]

R. Quintanilla, On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures, Acta Mechanica, 168 (2004), 61-73.  doi: 10.1007/s00707-004-0073-6.

[26]

R. Quintanilla and R. Racke, Qualitative aspects of solutions in resonators, Arch. Mech., 60 (2008), 345-360. 

[27]

R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-435. 

[28]

R. Racke, Thermoelasticity, Chapter 4 in: Handbook of Differential Equations, Evolutionary Equations. Eds.: C.M. Dafermos, M. Pokorný. Elsevier, 5 (2009), 315-420.

[29]

R. Racke, Heat conduction in elastic systems: Fourier versus Cattaneo, Proc. International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Skukuza, South Africa, (2015), 356-360. 

[30]

R. Racke and Y. Ueda, Dissipative structures for thermoelastic plate equations in $\mathbf{R}^n$, Advances. Differential Equations, 21 (2016), 601-630. 

[31]

X. Yang, Generalized Form of Hurwitz-Routh criterion and Hopf bifurcation of higher order, Appl. Math. Letters, 15 (2002), 615-621.  doi: 10.1016/S0893-9659(02)80014-3.

[32]

H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.  doi: 10.1093/imamat/hxh101.

show all references

References:
[1]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Instit. Mat. Univ. Trieste Suppl., 28 (1996), 1-28. 

[2]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. 

[3]

P. J. ChenM. E. Gurtin and W. O. Williams, A note on non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970.  doi: 10.1007/BF01602278.

[4]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112. 

[5]

R. Denk and R. Racke, $L^p$ resolvent estimates and time decay for generalized thermoelastic plate equations, Electronic J. Differential Equations, 48 (2006), 1-16. 

[6]

R. DenkR. Racke and Y. Shibata, $L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains, Advances Differential Equations, 14 (2009), 685-715. 

[7]

R. DenkR. Racke and Y. Shibata, Local energy decay estimate of solutions to the thermoelastic plate equations in two-and three-dimensional exterior domains, J. Analysis Appl., 29 (2010), 21-62.  doi: 10.4171/ZAA/1396.

[8]

H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound, J. Math. Phys. , 53 (2012), 073509, 13 pp. doi: 10.1063/1.4734239.

[9]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems --Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.

[10]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394.  doi: 10.1090/S0002-9947-1978-0461206-1.

[11]

F. Huang, Characteristic conditions for exponential staility of linear dynamical systems in Hilbert spaces, Ann. Diff. Equations, 1 (1985), 43-56. 

[12]

B. Kabil, Zur Asymptotik bei Resonator-Gleichungen, Diplomarbeit (diploma thesis), University of Konstanz, 2011.

[13]

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.

[14]

I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermoelastic equations, Advances Differential Equations, 3 (1998), 387-416. 

[15]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM, Proc., 4 (1998), 199-222.  doi: 10.1051/proc:1998029.

[16]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions, Abstract Appl. Anal., 3 (1998), 153-169.  doi: 10.1155/S1085337598000487.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457-482. 

[18]

M. C. LeseduarteR. Quintanilla and R. Racke, On non-exponential decay in generalized thermoelasticity with two temperatures, Appl. Math. Letters, 70 (2017), 18-25.  doi: 10.1016/j.aml.2017.02.020.

[19]

K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. angew. Math. Phys., 48 (1997), 885-904.  doi: 10.1007/s000330050071.

[20]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Res. Notes Math., 1999.

[21]

J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058.

[22]

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.

[23]

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

[24]

R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures, Dynamics Continous, Discrete Impulsive Sys., Ser. A: Math. Anal., 11 (2004), 57-68. 

[25]

R. Quintanilla, On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures, Acta Mechanica, 168 (2004), 61-73.  doi: 10.1007/s00707-004-0073-6.

[26]

R. Quintanilla and R. Racke, Qualitative aspects of solutions in resonators, Arch. Mech., 60 (2008), 345-360. 

[27]

R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-435. 

[28]

R. Racke, Thermoelasticity, Chapter 4 in: Handbook of Differential Equations, Evolutionary Equations. Eds.: C.M. Dafermos, M. Pokorný. Elsevier, 5 (2009), 315-420.

[29]

R. Racke, Heat conduction in elastic systems: Fourier versus Cattaneo, Proc. International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Skukuza, South Africa, (2015), 356-360. 

[30]

R. Racke and Y. Ueda, Dissipative structures for thermoelastic plate equations in $\mathbf{R}^n$, Advances. Differential Equations, 21 (2016), 601-630. 

[31]

X. Yang, Generalized Form of Hurwitz-Routh criterion and Hopf bifurcation of higher order, Appl. Math. Letters, 15 (2002), 615-621.  doi: 10.1016/S0893-9659(02)80014-3.

[32]

H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.  doi: 10.1093/imamat/hxh101.

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