American Institute of Mathematical Sciences

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May  2021, 26(5): 2805-2828. doi: 10.3934/dcdsb.2020206

Global and exponential attractors for a nonlinear porous elastic system with delay term

Manoel J. Dos Santos 1,, , Baowei Feng 2, , Dilberto S. Almeida Júnior 3, and  Mauro L. Santos 3,

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

* Corresponding author: Manoel J. Dos Santos

Received  July 2019 Revised  May 2020 Published  May 2021 Early access  July 2020

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.

Keywords: Nonlinear porous-elastic system, Time delay, Quasi-stability, Global attractor, Exponential attractors.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35L53; Secondary: 74K10, 93D20.
Citation: Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
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]

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.

[1]

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