# American Institute of Mathematical Sciences

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 & 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.   Google Scholar [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.   Google Scholar [3] P. J. Chen, M. 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.  Google Scholar [4] P. J. Chen, M. 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.   Google Scholar [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.   Google Scholar [6] R. Denk, R. Racke and Y. 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Soc., 236 (1978), 385-394.  doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar [11] F. Huang, Characteristic conditions for exponential staility of linear dynamical systems in Hilbert spaces, Ann. Diff. Equations, 1 (1985), 43-56.   Google Scholar [12] B. Kabil, Zur Asymptotik bei Resonator-Gleichungen, Diplomarbeit (diploma thesis), University of Konstanz, 2011. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [17] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457-482.   Google Scholar [18] M. C. Leseduarte, R. 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.  Google Scholar [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.  Google Scholar [20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Res. Notes Math., 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] J. Prüß, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar [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.   Google Scholar [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.  Google Scholar [26] R. Quintanilla and R. Racke, Qualitative aspects of solutions in resonators, Arch. Mech., 60 (2008), 345-360.   Google Scholar [27] R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-435.   Google Scholar [28] R. Racke, Thermoelasticity, Chapter 4 in: Handbook of Differential Equations, Evolutionary Equations. Eds.: C.M. Dafermos, M. Pokorný. Elsevier, 5 (2009), 315-420. Google Scholar [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.   Google Scholar [30] R. Racke and Y. Ueda, Dissipative structures for thermoelastic plate equations in $\mathbf{R}^n$, Advances. Differential Equations, 21 (2016), 601-630.   Google Scholar [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.  Google Scholar [32] H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.  doi: 10.1093/imamat/hxh101.  Google Scholar

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.   Google Scholar [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.   Google Scholar [3] P. J. Chen, M. 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.  Google Scholar [4] P. J. Chen, M. 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.   Google Scholar [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.   Google Scholar [6] R. Denk, R. 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.   Google Scholar [7] R. Denk, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] F. Huang, Characteristic conditions for exponential staility of linear dynamical systems in Hilbert spaces, Ann. Diff. Equations, 1 (1985), 43-56.   Google Scholar [12] B. Kabil, Zur Asymptotik bei Resonator-Gleichungen, Diplomarbeit (diploma thesis), University of Konstanz, 2011. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [17] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457-482.   Google Scholar [18] M. C. Leseduarte, R. 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.  Google Scholar [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.  Google Scholar [20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Res. Notes Math., 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] J. Prüß, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar [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.   Google Scholar [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.  Google Scholar [26] R. Quintanilla and R. Racke, Qualitative aspects of solutions in resonators, Arch. Mech., 60 (2008), 345-360.   Google Scholar [27] R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-435.   Google Scholar [28] R. Racke, Thermoelasticity, Chapter 4 in: Handbook of Differential Equations, Evolutionary Equations. Eds.: C.M. Dafermos, M. Pokorný. Elsevier, 5 (2009), 315-420. Google Scholar [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.   Google Scholar [30] R. Racke and Y. Ueda, Dissipative structures for thermoelastic plate equations in $\mathbf{R}^n$, Advances. Differential Equations, 21 (2016), 601-630.   Google Scholar [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.  Google Scholar [32] H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.  doi: 10.1093/imamat/hxh101.  Google Scholar
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