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Global and exponential attractors for a nonlinear porous elastic system with delay term
1. | Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu St., Abaetetuba, Pará, 68440-000, Brazil |
2. | Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China |
3. | PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa St., 01, 66075-110, Belém - Pará - Brazil |
This paper is concerned with the study on the existence of attractors for a nonlinear porous elastic system subjected to a delay-type damping in the volume fraction equation. The study will be performed, from the point of view of quasi-stability for infinite dimensional dynamical systems and from then on we will have the result of the existence of global and exponential attractors.
References:
[1] |
K. Ammari and S. Gerbi,
Interior feedback stabilization of wave equations with dynamic boundary delay, Z. Anal. Anwend, 36 (2017), 297-327.
doi: 10.4171/ZAA/1590. |
[2] |
M. Aouadi, Long-time dynamics for nonlinear porous thermoelasticity with second sound and delay, Journal of Mathematical Physics, 59 (2018), 101510, 23pp.
doi: 10.1063/1.5044615. |
[3] |
T. Apalara, Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay, Elect. J. Diff. Equ., 2014 (2014), 1–15.https://ejde.math.txstate.edu/Volumes/2014/254/apalara.pdf |
[4] |
T. Apalara,
Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Applicable Analysis, 95 (2016), 187-202.
doi: 10.1080/00036811.2014.1000314. |
[5] |
T. Apalara,
General decay of solutions in one-dimensional porous-elastic system with memory, Journal of Mathematical Analysis and Applications, 469 (2019), 457-471.
doi: 10.1016/j.jmaa.2017.08.007. |
[6] |
T. Apalara and S. Messaoudi,
An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Applied Mathematics and Optimization, 71 (2014), 449-472.
doi: 10.1007/s00245-014-9266-0. |
[7] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, Elsevier Science, 1992. |
[8] |
A. Barbosa and T. Ma,
Long-time dynamics of an extensible plate equation with thermal memory, Journal of Mathematical Analysis and Applications, 416 (2014), 143-165.
doi: 10.1016/j.jmaa.2014.02.042. |
[9] |
A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Elect. J. Diff. Equ., 2014 (2014), 1–11. https://ejde.math.txstate.edu/Volumes/2014/212/benseghir.pdf |
[10] |
P. S. Casas and R. Quintanilla,
Exponential decay in one-dimensional porous-thermoelasticity, Mechanics Research Communications, 32 (2005), 652-658.
doi: 10.1016/j.mechrescom.2005.02.015. |
[11] |
I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, https://www.emis.de/monographs/Chueshov/book.pdf |
[12] |
I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the American Mathematical Society, 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[13] |
I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, 2010.
doi: 10.1007/978-0-387-87712-9. |
[14] |
S. C. Cowin and J. W. Nunziato,
Linear elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147.
doi: 10.1007/BF00041230. |
[15] |
Q. Dai and Z. Yang,
Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Angew. Math. Phys., 65 (2014), 885-903.
doi: 10.1007/s00033-013-0365-6. |
[16] |
R. Datko,
Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Jounal on Control and Optimization, 26 (1988), 697-713.
doi: 10.1137/0326040. |
[17] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[18] |
L. H. Fatori, M. A. J. Silva and V. Narciso,
Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete & Continuous Dynamical Systems–A, 36 (2016), 6117-6132.
doi: 10.3934/dcds.2016067. |
[19] |
B. Feng and M. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay, Boundary Value Problems, 2015 (2015), 13pp.
doi: 10.1186/s13661-015-0468-4. |
[20] |
B. Feng and X. Yang,
Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.
doi: 10.1080/00036811.2016.1148139. |
[21] |
B. Feng and M. Yin,
Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds, Mathematics and Mechanics of Solids, 24 (2019), 2361-2373.
doi: 10.1177/1081286518757299. |
[22] |
M. M. Freitas, M. L. Santos and L. A. Langa,
Porous elastic system with nonlinear damping and sources terms, Journal of Differential Equations, 264 (2018), 2970-3051.
doi: 10.1016/j.jde.2017.11.006. |
[23] |
E. Friedman, S. Nicaise and S. Valein,
Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM Journal on Control and Optimization, 48 (2010), 5028-5052.
doi: 10.1137/090762105. |
[24] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, American Mathematical Society, 1988. https://books.google.com.br/books/about/Asymptotic_Behavior_of_Dissipative_Syste.html?id=3DuNyCB294cC&redir_esc=y |
[25] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied mathematical sciences, Springer-Verlag, 1993. https://books.google.com.br/books?id=DVsZAQAAIAAJ.
doi: 10.1007/978-1-4612-4342-7. |
[26] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliae Mathematica, 46 (1989), 245–258. http://purl.pt/3178 |
[27] |
D. Iesan, Thermoelastic Models of Continua, Springer Netherlands, 2004.
doi: 10.1007/978-1-4020-2310-1. |
[28] |
M. Kirane and B. Said-Houari,
Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.
doi: 10.1007/s00033-011-0145-0. |
[29] |
M. C. Leseduarte, A. Magaña and R. Quintanilla,
On the time decay of solutions in porous-thermo-elasticity of type Ⅱ, Discrete & Continuous Dynamical Systems - B, 13 (2010), 375-391.
doi: 10.3934/dcdsb.2010.13.375. |
[30] |
G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Elect. J. Diff. Equ., 2017 (2017), 1–13. https://ejde.math.txstate.edu/Volumes/2017/174/liu.pdf |
[31] |
K. Liu,
Locally distributed control and damping for the conservative systems, SIAM Journal on Control and Optimization, 35 (1997), 1574-1590.
doi: 10.1137/S0363012995284928. |
[32] |
W. Liu and M. Chen,
Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mech. Thermodyn, 29 (2017), 731-746.
doi: 10.1007/s00161-017-0556-z. |
[33] |
W. J. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys., 54 (2013), 043504, 9pp.
doi: 10.1063/1.4799929. |
[34] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999. https://books.google.com.br/books/about/Semigroups_Associated_with_Dissipative_S.html?id=ReG5eHHshpoC&redir_esc=y |
[35] |
T. F. Ma and R. Monteiro,
Singular limit and long-time dynamics of bresse systems, SIAM Journal on Mathematical Analysis, 49 (2017), 2468-2495.
doi: 10.1137/15M1039894. |
[36] |
A. Magaña and R. Quintanilla,
On the spatial behavior of solutions for porous elastic solids with quasi-static microvoids, Mathematical and Computer Modelling, 44 (2006), 710-716.
doi: 10.1016/j.mcm.2006.02.007. |
[37] |
J. Muñoz-Rivera and R. Quintanilla,
On the time polynomial decay in elastic solids with voids, Journal of Mathematical Analysis and Applications, 338 (2008), 1296-1309.
doi: 10.1016/j.jmaa.2007.06.005. |
[38] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[39] |
S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Int. Equ., 21 (2008), 935–958. https://projecteuclid.org/euclid.die/1356038593 |
[40] |
S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 2011 (2011), 1–20. https://ejde.math.txstate.edu/Volumes/2011/41/nicaise.pdf |
[41] |
S. Nicaise, J. Valein and E. Fridman,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems–S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
[42] |
J. W. Nunziato and S. C. Cowin,
A nonlinear theory of elastic materials with voids, Arquive for Rational Mechanical Analysis, 72 (1979), 175-201.
doi: 10.1007/BF00249363. |
[43] |
H. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[44] |
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. |
[45] |
C. A. Raposo, T. A. Apalara and R. J. Ribeiro,
Analyticity to transmission problem with delay in porous-elasticity, Journal of Mathematical Analysis and Applications, 466 (2018), 819-834.
doi: 10.1016/j.jmaa.2018.06.017. |
[46] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.
doi: 10.1115/1.1579456.![]() ![]() ![]() |
[47] |
B. Said-Houari and Y. Laskri,
A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.
doi: 10.1016/j.amc.2010.08.021. |
[48] |
M. L. Santos and D. S. Almeida Júnior,
On the porous-elastic system with kelvin–voigt damping, Journal of Mathematical Analysis and Applications, 445 (2017), 498-512.
doi: 10.1016/j.jmaa.2016.08.005. |
[49] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
On the decay rates of porous elastic systems, Journal of Elasticity, 127 (2017), 79-101.
doi: 10.1007/s10659-016-9597-y. |
[50] |
M. L. Santos, A. D. S. Campelo and M. L. S. Oliveira,
On porous-elastic systems with fourier law, Applicable Analysis, 98 (2019), 1181-1197.
doi: 10.1080/00036811.2017.1419197. |
[51] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Springer New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[52] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-0645-3. |
[53] |
D. Wang, G. Li and B. Zhu, Exponential energy decay of solutions for a transmission problem with viscoelastic term and delay, Mathematics, 4 (2016), 42.
doi: 10.3390/math4020042. |
[54] |
C. Q. Xu, S. P. Yung and K. L. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[55] |
X. Yang, J. Zhang and Y. Lu, Dynamics of the Nonlinear Timoshenko System with Variable Delay, Applied Mathematics & Optimization, 2018.
doi: 10.1007/s00245-018-9539-0. |
[56] |
E. Zuazua,
Exponential decay for the semilinear wave equation with locally distributed damping, Communications in Partial Differential Equations, 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
show all references
References:
[1] |
K. Ammari and S. Gerbi,
Interior feedback stabilization of wave equations with dynamic boundary delay, Z. Anal. Anwend, 36 (2017), 297-327.
doi: 10.4171/ZAA/1590. |
[2] |
M. Aouadi, Long-time dynamics for nonlinear porous thermoelasticity with second sound and delay, Journal of Mathematical Physics, 59 (2018), 101510, 23pp.
doi: 10.1063/1.5044615. |
[3] |
T. Apalara, Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay, Elect. J. Diff. Equ., 2014 (2014), 1–15.https://ejde.math.txstate.edu/Volumes/2014/254/apalara.pdf |
[4] |
T. Apalara,
Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Applicable Analysis, 95 (2016), 187-202.
doi: 10.1080/00036811.2014.1000314. |
[5] |
T. Apalara,
General decay of solutions in one-dimensional porous-elastic system with memory, Journal of Mathematical Analysis and Applications, 469 (2019), 457-471.
doi: 10.1016/j.jmaa.2017.08.007. |
[6] |
T. Apalara and S. Messaoudi,
An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Applied Mathematics and Optimization, 71 (2014), 449-472.
doi: 10.1007/s00245-014-9266-0. |
[7] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, Elsevier Science, 1992. |
[8] |
A. Barbosa and T. Ma,
Long-time dynamics of an extensible plate equation with thermal memory, Journal of Mathematical Analysis and Applications, 416 (2014), 143-165.
doi: 10.1016/j.jmaa.2014.02.042. |
[9] |
A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Elect. J. Diff. Equ., 2014 (2014), 1–11. https://ejde.math.txstate.edu/Volumes/2014/212/benseghir.pdf |
[10] |
P. S. Casas and R. Quintanilla,
Exponential decay in one-dimensional porous-thermoelasticity, Mechanics Research Communications, 32 (2005), 652-658.
doi: 10.1016/j.mechrescom.2005.02.015. |
[11] |
I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, https://www.emis.de/monographs/Chueshov/book.pdf |
[12] |
I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the American Mathematical Society, 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[13] |
I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, 2010.
doi: 10.1007/978-0-387-87712-9. |
[14] |
S. C. Cowin and J. W. Nunziato,
Linear elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147.
doi: 10.1007/BF00041230. |
[15] |
Q. Dai and Z. Yang,
Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Angew. Math. Phys., 65 (2014), 885-903.
doi: 10.1007/s00033-013-0365-6. |
[16] |
R. Datko,
Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Jounal on Control and Optimization, 26 (1988), 697-713.
doi: 10.1137/0326040. |
[17] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[18] |
L. H. Fatori, M. A. J. Silva and V. Narciso,
Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete & Continuous Dynamical Systems–A, 36 (2016), 6117-6132.
doi: 10.3934/dcds.2016067. |
[19] |
B. Feng and M. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay, Boundary Value Problems, 2015 (2015), 13pp.
doi: 10.1186/s13661-015-0468-4. |
[20] |
B. Feng and X. Yang,
Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.
doi: 10.1080/00036811.2016.1148139. |
[21] |
B. Feng and M. Yin,
Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds, Mathematics and Mechanics of Solids, 24 (2019), 2361-2373.
doi: 10.1177/1081286518757299. |
[22] |
M. M. Freitas, M. L. Santos and L. A. Langa,
Porous elastic system with nonlinear damping and sources terms, Journal of Differential Equations, 264 (2018), 2970-3051.
doi: 10.1016/j.jde.2017.11.006. |
[23] |
E. Friedman, S. Nicaise and S. Valein,
Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM Journal on Control and Optimization, 48 (2010), 5028-5052.
doi: 10.1137/090762105. |
[24] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, American Mathematical Society, 1988. https://books.google.com.br/books/about/Asymptotic_Behavior_of_Dissipative_Syste.html?id=3DuNyCB294cC&redir_esc=y |
[25] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied mathematical sciences, Springer-Verlag, 1993. https://books.google.com.br/books?id=DVsZAQAAIAAJ.
doi: 10.1007/978-1-4612-4342-7. |
[26] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliae Mathematica, 46 (1989), 245–258. http://purl.pt/3178 |
[27] |
D. Iesan, Thermoelastic Models of Continua, Springer Netherlands, 2004.
doi: 10.1007/978-1-4020-2310-1. |
[28] |
M. Kirane and B. Said-Houari,
Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.
doi: 10.1007/s00033-011-0145-0. |
[29] |
M. C. Leseduarte, A. Magaña and R. Quintanilla,
On the time decay of solutions in porous-thermo-elasticity of type Ⅱ, Discrete & Continuous Dynamical Systems - B, 13 (2010), 375-391.
doi: 10.3934/dcdsb.2010.13.375. |
[30] |
G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Elect. J. Diff. Equ., 2017 (2017), 1–13. https://ejde.math.txstate.edu/Volumes/2017/174/liu.pdf |
[31] |
K. Liu,
Locally distributed control and damping for the conservative systems, SIAM Journal on Control and Optimization, 35 (1997), 1574-1590.
doi: 10.1137/S0363012995284928. |
[32] |
W. Liu and M. Chen,
Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mech. Thermodyn, 29 (2017), 731-746.
doi: 10.1007/s00161-017-0556-z. |
[33] |
W. J. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys., 54 (2013), 043504, 9pp.
doi: 10.1063/1.4799929. |
[34] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999. https://books.google.com.br/books/about/Semigroups_Associated_with_Dissipative_S.html?id=ReG5eHHshpoC&redir_esc=y |
[35] |
T. F. Ma and R. Monteiro,
Singular limit and long-time dynamics of bresse systems, SIAM Journal on Mathematical Analysis, 49 (2017), 2468-2495.
doi: 10.1137/15M1039894. |
[36] |
A. Magaña and R. Quintanilla,
On the spatial behavior of solutions for porous elastic solids with quasi-static microvoids, Mathematical and Computer Modelling, 44 (2006), 710-716.
doi: 10.1016/j.mcm.2006.02.007. |
[37] |
J. Muñoz-Rivera and R. Quintanilla,
On the time polynomial decay in elastic solids with voids, Journal of Mathematical Analysis and Applications, 338 (2008), 1296-1309.
doi: 10.1016/j.jmaa.2007.06.005. |
[38] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[39] |
S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Int. Equ., 21 (2008), 935–958. https://projecteuclid.org/euclid.die/1356038593 |
[40] |
S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 2011 (2011), 1–20. https://ejde.math.txstate.edu/Volumes/2011/41/nicaise.pdf |
[41] |
S. Nicaise, J. Valein and E. Fridman,
Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems–S, 2 (2009), 559-581.
doi: 10.3934/dcdss.2009.2.559. |
[42] |
J. W. Nunziato and S. C. Cowin,
A nonlinear theory of elastic materials with voids, Arquive for Rational Mechanical Analysis, 72 (1979), 175-201.
doi: 10.1007/BF00249363. |
[43] |
H. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[44] |
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. |
[45] |
C. A. Raposo, T. A. Apalara and R. J. Ribeiro,
Analyticity to transmission problem with delay in porous-elasticity, Journal of Mathematical Analysis and Applications, 466 (2018), 819-834.
doi: 10.1016/j.jmaa.2018.06.017. |
[46] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.
doi: 10.1115/1.1579456.![]() ![]() ![]() |
[47] |
B. Said-Houari and Y. Laskri,
A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.
doi: 10.1016/j.amc.2010.08.021. |
[48] |
M. L. Santos and D. S. Almeida Júnior,
On the porous-elastic system with kelvin–voigt damping, Journal of Mathematical Analysis and Applications, 445 (2017), 498-512.
doi: 10.1016/j.jmaa.2016.08.005. |
[49] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
On the decay rates of porous elastic systems, Journal of Elasticity, 127 (2017), 79-101.
doi: 10.1007/s10659-016-9597-y. |
[50] |
M. L. Santos, A. D. S. Campelo and M. L. S. Oliveira,
On porous-elastic systems with fourier law, Applicable Analysis, 98 (2019), 1181-1197.
doi: 10.1080/00036811.2017.1419197. |
[51] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Springer New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[52] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-0645-3. |
[53] |
D. Wang, G. Li and B. Zhu, Exponential energy decay of solutions for a transmission problem with viscoelastic term and delay, Mathematics, 4 (2016), 42.
doi: 10.3390/math4020042. |
[54] |
C. Q. Xu, S. P. Yung and K. L. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[55] |
X. Yang, J. Zhang and Y. Lu, Dynamics of the Nonlinear Timoshenko System with Variable Delay, Applied Mathematics & Optimization, 2018.
doi: 10.1007/s00245-018-9539-0. |
[56] |
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