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Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping
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 |
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.
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. |
[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.
|
[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. |
[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. |
[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. |
[6] |
R. M. Bowen, Continuum Physics III: Theory of Mixtures, A. C. Eringen, ed., Academic Press, New York, (1976), 689–722. |
[7] |
R. M. Bowen and J. C. Wiese,
Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc, 2011.
doi: 10.1090/surv/176. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
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. |
[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.
|
[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. |
[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. |
[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. |
[6] |
R. M. Bowen, Continuum Physics III: Theory of Mixtures, A. C. Eringen, ed., Academic Press, New York, (1976), 689–722. |
[7] |
R. M. Bowen and J. C. Wiese,
Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc, 2011.
doi: 10.1090/surv/176. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
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