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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

Zhong-Jie Han 1, and  Gen-Qi Xu 1,

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.
Keywords: Porous-thermo-elasticity, spectrum-determined-growth condition, semigroup, exponential stability., non-uniform, spectrum.
Mathematics Subject Classification: Primary: 35B40, 35P20; Secondary: 74F0.
Citation: Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[7]

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

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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