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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

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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

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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

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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

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