American Institute of Mathematical Sciences

Advanced
February  2020, 19(2): 785-809. doi: 10.3934/cpaa.2020037

Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping

Mirelson M. Freitas 1, , Alberto L. C. Costa 2, and  Geraldo M. Araújo 2,

1. 

Federal University of Pará, Raimundo Santana Street s/n, Salinópolis PA, 68721-000, Brazil
2. 

Institute of Exact and Natural Sciences, Doctoral Program in Mathematics, Federal University of Pará, Augusto corrêa Street, Number 01, 66075-110, Belém PA, Brazil

Received  November 2018 Revised  March 2019 Published  October 2019

This paper is devoted to study the asymptotic behavior of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. We prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Moreover, we prove the upper-semicontinuity of pullback attractors with respect to non-autonomous perturbations.

Keywords: Mixture problem, nonlinear damping, pullback attractors, non-autonomous dynamical systems, upper-semicontinuity.
Mathematics Subject Classification: Primary: 35B40, 35L53, 35B41; Secondary: 37B55.
Citation: Mirelson M. Freitas, Alberto L. C. Costa, Geraldo M. Araújo. Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. Communications on Pure & Applied Analysis, 2020, 19 (2) : 785-809. doi: 10.3934/cpaa.2020037
References:
[1]

M. S. Alves, J. E. Muñoz Rivera and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids, Internat J. Solids Struct., 46 (2009), 1659-1666.  doi: 10.1016/j.ijsolstr.2008.12.005.  Google Scholar

[2]

M. S. Alves, J. E. Muñoz Rivera, M. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat J. Solids Struct., 46 (2009), 4151-4162.   Google Scholar

[3]

R. J. Atkin and R. E. Craine, Continuum theories of mixtures: basic theory and hystorical development, Quat. J. Mech. Appl. Math., 29 (1976), 209-243.  doi: 10.1093/qjmam/29.2.209.  Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, volume 190 of Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[5]

A. Bedford and D. S. Drumheller, Theory of immiscible and structured mixtures, Int. J. Eng. Sci., 21 (1983), 863-960.  doi: 10.1016/0020-7225(83)90071-X.  Google Scholar

[6]

R. M. Bowen, Continuum Physics III: Theory of Mixtures, A. C. Eringen, ed., Academic Press, New York, (1976), 689–722.  Google Scholar

[7]

R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.   Google Scholar

[8]

T. Caraballo, G. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, vol. 195, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of the AMS 195, Providence, RI, 2008. doi: 10.1090/memo/0912.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[13]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4353.  doi: 10.1016/j.jde.2012.01.010.  Google Scholar

[14]

D. Iesan and R. Quintanilla, Existence and continuous dependence results in the theory of interacting continua, J. Elasticity, 36 (1994), 85-98.  doi: 10.1007/BF00042493.  Google Scholar

[15]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc, 2011. doi: 10.1090/surv/176.  Google Scholar

[16]

T. F. Ma, P. Marín-Rubio and M. S. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

[17]

T. F. Ma and R. N. 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

[18]

T. F. Ma and T. M. Souza, Pullback dynamics of non-autonomous wave equations with acoustic boundary condition, Differential and Integral Equations, 30 (2017), 443-462.   Google Scholar

[19]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3956-3963.  doi: 10.1016/j.na.2009.02.065.  Google Scholar

[20]

F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids, Collect. Math., 46 (1995), 236-277.   Google Scholar

[21]

M. L. Santos and M. M. Freitas, Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms, Comm. Pure Appl. Anal., 18 (2019), 1869-1890.   Google Scholar

[22]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[23]

C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical non-linearity, Nonlinearity, 19 (2006), 2645-2665.  doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[24]

Y. Wang, On the upper semicontinuity of pullback attractors with applications to plate equations, Comm. Pure Appl. Anal., 9 (2010), 1653-1673.  doi: 10.3934/cpaa.2010.9.1653.  Google Scholar

show all references

References:
[1]

M. S. Alves, J. E. Muñoz Rivera and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids, Internat J. Solids Struct., 46 (2009), 1659-1666.  doi: 10.1016/j.ijsolstr.2008.12.005.  Google Scholar

[2]

M. S. Alves, J. E. Muñoz Rivera, M. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat J. Solids Struct., 46 (2009), 4151-4162.   Google Scholar

[3]

R. J. Atkin and R. E. Craine, Continuum theories of mixtures: basic theory and hystorical development, Quat. J. Mech. Appl. Math., 29 (1976), 209-243.  doi: 10.1093/qjmam/29.2.209.  Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, volume 190 of Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[5]

A. Bedford and D. S. Drumheller, Theory of immiscible and structured mixtures, Int. J. Eng. Sci., 21 (1983), 863-960.  doi: 10.1016/0020-7225(83)90071-X.  Google Scholar

[6]

R. M. Bowen, Continuum Physics III: Theory of Mixtures, A. C. Eringen, ed., Academic Press, New York, (1976), 689–722.  Google Scholar

[7]

R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.   Google Scholar

[8]

T. Caraballo, G. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, vol. 195, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of the AMS 195, Providence, RI, 2008. doi: 10.1090/memo/0912.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[13]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4353.  doi: 10.1016/j.jde.2012.01.010.  Google Scholar

[14]

D. Iesan and R. Quintanilla, Existence and continuous dependence results in the theory of interacting continua, J. Elasticity, 36 (1994), 85-98.  doi: 10.1007/BF00042493.  Google Scholar

[15]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc, 2011. doi: 10.1090/surv/176.  Google Scholar

[16]

T. F. Ma, P. Marín-Rubio and M. S. Surco Chuño, Dynamics of wave equations with moving boundary, J. Differential Equations, 262 (2017), 3317-3342.  doi: 10.1016/j.jde.2016.11.030.  Google Scholar

[17]

T. F. Ma and R. N. 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

[18]

T. F. Ma and T. M. Souza, Pullback dynamics of non-autonomous wave equations with acoustic boundary condition, Differential and Integral Equations, 30 (2017), 443-462.   Google Scholar

[19]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3956-3963.  doi: 10.1016/j.na.2009.02.065.  Google Scholar

[20]

F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids, Collect. Math., 46 (1995), 236-277.   Google Scholar

[21]

M. L. Santos and M. M. Freitas, Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms, Comm. Pure Appl. Anal., 18 (2019), 1869-1890.   Google Scholar

[22]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[23]

C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical non-linearity, Nonlinearity, 19 (2006), 2645-2665.  doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[24]

Y. Wang, On the upper semicontinuity of pullback attractors with applications to plate equations, Comm. Pure Appl. Anal., 9 (2010), 1653-1673.  doi: 10.3934/cpaa.2010.9.1653.  Google Scholar

[1]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036
[2]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
[3]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
[4]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[5]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
[6]

Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
[7]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214
[8]

Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259
[9]

Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279
[10]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[11]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[12]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[13]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653
[14]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[15]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
[16]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543
[17]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[18]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[19]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935
[20]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (49)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences