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Linear programming based Lyapunov function computation for differential inclusions
Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type
1. | Department of Mathematics, Tianjin University, Tianjin 300072 |
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. |
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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