July  2022, 27(7): 3563-3583. doi: 10.3934/dcdsb.2021196

Existence and continuity of global attractors for ternary mixtures of solids

1. 

Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, S/N, 68721-000, Salinópolis, PA, Brazil

2. 

Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

3. 

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

* Corresponding author: Mirelson M. Freitas

Received  December 2020 Revised  June 2021 Published  July 2022 Early access  August 2021

In this paper, we study the long-time dynamics of a system modelinga mixture of three interacting continua with nonlinear damping, sources terms and subjected to small perturbations of autonomousexternal forces with a parameter $ \epsilon $, inspired by the modelstudied by Dell' Oro and Rivera [12]. We establish astabilizability estimate for the associated gradient dynamicalsystem, which as a consequence, implies the existence of a compactglobal attractor with finite fractal dimension andexponential attractors. This estimate is establishedindependent of the parameter $ \epsilon\in[0,1] $. We also prove thesmoothness of global attractors independent of the parameter$ \epsilon\in[0,1] $. Moreover, we show that the family of globalattractors is continuous with respect to the parameter $ \epsilon $ ona residual dense set $ I_*\subset[0,1] $ in the same sense proposed inHoang et al. [15].

Citation: Mirelson M. Freitas, Anderson J. A. Ramos, Baowei Feng, Mauro L. Santos, Helen C. M. Rodrigues. Existence and continuity of global attractors for ternary mixtures of solids. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3563-3583. doi: 10.3934/dcdsb.2021196
References:
[1]

M. S. AlvesJ. 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. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat. J. Solids Struct., 46 (2009), 4151-4162.  doi: 10.1016/j.ijsolstr.2009.07.026.

[3]

R. J. Atkin and R. E. Craine, Continuum theories of mixtures: Basic theory and historical development, Quat. J. Mech. Appl. Math., 29 (1976), 209-244.  doi: 10.1093/qjmam/29.2.209.

[4]

A. V. Babin and S. Y. Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.

[5]

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

[6]

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.

[7]

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

[8]

R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.  doi: 10.1016/0020-7225(69)90048-2.

[9]

I. ChueshovM. Eller and 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.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear, damping, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912.

[11]

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

[12]

F. Dell' Oro and J. E. Muñoz Rivera, Stabilization of ternary mixtures with frictional dissipation, Asymptotic Analysis, 89 (2014), 235-262.  doi: 10.3233/ASY-141229.

[13]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[14]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singulary perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[15]

L. T. HoangE. J. Olson and and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc., 143 (2015), 4389-4395.  doi: 10.1090/proc/12598.

[16]

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.

[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]

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

[19]

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.

show all references

References:
[1]

M. S. AlvesJ. 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. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat. J. Solids Struct., 46 (2009), 4151-4162.  doi: 10.1016/j.ijsolstr.2009.07.026.

[3]

R. J. Atkin and R. E. Craine, Continuum theories of mixtures: Basic theory and historical development, Quat. J. Mech. Appl. Math., 29 (1976), 209-244.  doi: 10.1093/qjmam/29.2.209.

[4]

A. V. Babin and S. Y. Pilyugin, Continuous dependence of attractors on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.

[5]

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

[6]

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.

[7]

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

[8]

R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689-722.  doi: 10.1016/0020-7225(69)90048-2.

[9]

I. ChueshovM. Eller and 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.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear, damping, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912.

[11]

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

[12]

F. Dell' Oro and J. E. Muñoz Rivera, Stabilization of ternary mixtures with frictional dissipation, Asymptotic Analysis, 89 (2014), 235-262.  doi: 10.3233/ASY-141229.

[13]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[14]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singulary perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[15]

L. T. HoangE. J. Olson and and J. C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc., 143 (2015), 4389-4395.  doi: 10.1090/proc/12598.

[16]

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.

[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]

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

[19]

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.

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