January  2012, 17(1): 57-77. doi: 10.3934/dcdsb.2012.17.57

Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type

1. 

Department of Mathematics, Tianjin University, Tianjin 300072

Received  September 2010 Revised  July 2011 Published  October 2011

The spectrum and asymptotic behavior of the non-uniform porous-thermo-elasticity of Lord-Shulman type is considered in this paper. It is shown that the corresponding system operator generates a $C_0$ semigroup of contractions in an appropriate Hilbert space setting. By a detailed spectral analysis, the asymptotic expressions of the spectrum of the system is gotten. Based on the spectral property, the Riesz basis property of the (generalized) eigenvectors is proved, which implies that the system satisfies the spectrum-determined-growth condition. Then the exponential stability of this system is deduced from the distribution of the spectrum.
Citation: Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 57-77. doi: 10.3934/dcdsb.2012.17.57
References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, Journal of Differential Equations, 194 (2003), 82-115.

[3]

P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous-thermo-elasticity, Mechanics Research Communications, 32 (2005), 652-658. doi: 10.1016/j.mechrescom.2005.02.015.

[4]

S. Chiriţă, M. Ciarletta and B. Straughan, Structural stability in porous elasticity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2593-2605.

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J. B. Conway, "Functions of One Complex Variable," 2nd edition, Graduate Texts in Mathematics, 11, Springer-Verlag, New York-Berlin, 1978.

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S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230.

[7]

S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191. doi: 10.1007/BF00041992.

[8]

Y. Du and G. Q. Xu, Exponetial stability of a system of linear Timoshenko type with boundary controls, J. Sys. Sci. & Math. Scis., 28 (2008), 554-575.

[9]

L. H. Fatori and J. E. Muñoz Rivera, Energy decay for hyperbolic thermoelastic systems of memory type, Quart. Appl. Math., 59 (2001), 441-458.

[10]

B.-Z. Guo and G.-Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, Journal of Functional Analysis, 231 (2006), 245-268.

[11]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds," Oxford Mathematical Monographs, Oxford University Press, Oxford, 2010.

[12]

B. Lazzari and R. Nibbi, On the influence of a dissipative boundary on the energy decay for a porous elastic solid, Mechanics Research Communications, 36 (2009), 581-586. doi: 10.1016/j.mechrescom.2009.01.010.

[13]

M. C. Leseduarte, A. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375.

[14]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[15]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433.

[16]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15 (1967), 299-309. doi: 10.1016/0022-5096(67)90024-5.

[17]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.

[18]

A. Magaña and R. Quintanilla, On the time decay of solution in one-dimensional theories of porous materials, International Journal of Solids and Structures, 43 (2006), 3414-3427. doi: 10.1016/j.ijsolstr.2005.06.077.

[19]

A. Magaña and R. Quintanilla, On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity, Asymptotic Analysis, 49 (2006), 183-187.

[20]

R. Mennicken and M. Möller, "Non-Self-Adjoint Boundary Eigenvalue Problem," North-Holland Mathematics Studies, 192, North-Holland Publishing Co., Amsterdam, 2003.

[21]

J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338 (2008), 1296-1309. doi: 10.1016/j.jmaa.2007.06.005.

[22]

W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch. Ration. Mech. Anal., 72 (): 175.  doi: 10.1007/BF00249363.

[23]

P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla, Stabilization in elastic solids with voids, J. Math. Anal. Appl., 350 (2009), 37-49. doi: 10.1016/j.jmaa.2008.09.026.

[24]

P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl., 379 (2011), 682-705. doi: 10.1016/j.jmaa.2011.01.045.

[25]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[26]

R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity, Applied Mathematics Letters, 16 (2003), 487-491. doi: 10.1016/S0893-9659(03)00025-9.

[27]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[28]

R. Racke and Y.-G. Wang, Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound, Zeitschrift fur Analysis und ihre Anwendungen, 24 (2005), 117-135. doi: 10.4171/ZAA/1232.

[29]

R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328.

[30]

A. Soufyane, M. Afilal and M. Chacha, Boundary stabilization of memory type for the porous-thermo-elasticity system,, Abstract and Applied Analysis, 2009 (). 

[31]

A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Applicable Analysis, 87 (2008), 451-464. doi: 10.1080/00036810802035634.

[32]

C. Tretter, Boundary eigenvalue problems for differential equations $N\eta=\lambda P\eta$ with $\lambda-$polynomial boundary conditions, Journal of Differential Equations, 170 (2001), 408-471.

[33]

J.-M. Wang and B.-Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type, Journal of the Franklin Institute, 344 (2007), 75-96. doi: 10.1016/j.jfranklin.2005.10.003.

[34]

G. Q. Xu and S. P. Yung, The expansion of semigroup and a Riesz basis criterion, Journal of Differential Equations, 210 (2005), 1-24.

[35]

G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams, International Journal of Control, 80 (2007), 470-485. doi: 10.1080/00207170601100904.

[36]

R. M. Young, "An Introduction to Nonharmonic Fourier Series," Pure and Applied Mathematics, 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, Journal of Differential Equations, 194 (2003), 82-115.

[3]

P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous-thermo-elasticity, Mechanics Research Communications, 32 (2005), 652-658. doi: 10.1016/j.mechrescom.2005.02.015.

[4]

S. Chiriţă, M. Ciarletta and B. Straughan, Structural stability in porous elasticity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2593-2605.

[5]

J. B. Conway, "Functions of One Complex Variable," 2nd edition, Graduate Texts in Mathematics, 11, Springer-Verlag, New York-Berlin, 1978.

[6]

S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230.

[7]

S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191. doi: 10.1007/BF00041992.

[8]

Y. Du and G. Q. Xu, Exponetial stability of a system of linear Timoshenko type with boundary controls, J. Sys. Sci. & Math. Scis., 28 (2008), 554-575.

[9]

L. H. Fatori and J. E. Muñoz Rivera, Energy decay for hyperbolic thermoelastic systems of memory type, Quart. Appl. Math., 59 (2001), 441-458.

[10]

B.-Z. Guo and G.-Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, Journal of Functional Analysis, 231 (2006), 245-268.

[11]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds," Oxford Mathematical Monographs, Oxford University Press, Oxford, 2010.

[12]

B. Lazzari and R. Nibbi, On the influence of a dissipative boundary on the energy decay for a porous elastic solid, Mechanics Research Communications, 36 (2009), 581-586. doi: 10.1016/j.mechrescom.2009.01.010.

[13]

M. C. Leseduarte, A. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375.

[14]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[15]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433.

[16]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15 (1967), 299-309. doi: 10.1016/0022-5096(67)90024-5.

[17]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.

[18]

A. Magaña and R. Quintanilla, On the time decay of solution in one-dimensional theories of porous materials, International Journal of Solids and Structures, 43 (2006), 3414-3427. doi: 10.1016/j.ijsolstr.2005.06.077.

[19]

A. Magaña and R. Quintanilla, On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity, Asymptotic Analysis, 49 (2006), 183-187.

[20]

R. Mennicken and M. Möller, "Non-Self-Adjoint Boundary Eigenvalue Problem," North-Holland Mathematics Studies, 192, North-Holland Publishing Co., Amsterdam, 2003.

[21]

J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338 (2008), 1296-1309. doi: 10.1016/j.jmaa.2007.06.005.

[22]

W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch. Ration. Mech. Anal., 72 (): 175.  doi: 10.1007/BF00249363.

[23]

P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla, Stabilization in elastic solids with voids, J. Math. Anal. Appl., 350 (2009), 37-49. doi: 10.1016/j.jmaa.2008.09.026.

[24]

P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl., 379 (2011), 682-705. doi: 10.1016/j.jmaa.2011.01.045.

[25]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[26]

R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity, Applied Mathematics Letters, 16 (2003), 487-491. doi: 10.1016/S0893-9659(03)00025-9.

[27]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[28]

R. Racke and Y.-G. Wang, Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound, Zeitschrift fur Analysis und ihre Anwendungen, 24 (2005), 117-135. doi: 10.4171/ZAA/1232.

[29]

R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328.

[30]

A. Soufyane, M. Afilal and M. Chacha, Boundary stabilization of memory type for the porous-thermo-elasticity system,, Abstract and Applied Analysis, 2009 (). 

[31]

A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Applicable Analysis, 87 (2008), 451-464. doi: 10.1080/00036810802035634.

[32]

C. Tretter, Boundary eigenvalue problems for differential equations $N\eta=\lambda P\eta$ with $\lambda-$polynomial boundary conditions, Journal of Differential Equations, 170 (2001), 408-471.

[33]

J.-M. Wang and B.-Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type, Journal of the Franklin Institute, 344 (2007), 75-96. doi: 10.1016/j.jfranklin.2005.10.003.

[34]

G. Q. Xu and S. P. Yung, The expansion of semigroup and a Riesz basis criterion, Journal of Differential Equations, 210 (2005), 1-24.

[35]

G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams, International Journal of Control, 80 (2007), 470-485. doi: 10.1080/00207170601100904.

[36]

R. M. Young, "An Introduction to Nonharmonic Fourier Series," Pure and Applied Mathematics, 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

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