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

Manoel J. Dos Santos; Baowei Feng; Dilberto S. Almeida Júnior; Mauro L. Santos

doi:10.3934/dcdsb.2020206

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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 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 & 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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, Elsevier Science, 1992. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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 Google Scholar
[27]	D. Iesan, Thermoelastic Models of Continua, Springer Netherlands, 2004. doi: 10.1007/978-1-4020-2310-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, Elsevier Science, 1992. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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 Google Scholar
[27]	D. Iesan, Thermoelastic Models of Continua, Springer Netherlands, 2004. doi: 10.1007/978-1-4020-2310-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 Google Scholar
[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 Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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

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

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