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July  2019, 18(4): 1869-1890. doi: 10.3934/cpaa.2019087

Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms

1. 

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

2. 

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

Received  July 2018 Revised  July 2018 Published  January 2019

Fund Project: M. L. Santos is supported by CNPq Grant 302899/2015-4 and by CNPq Grant 401769-0 (Universal Project-2016).

This paper is concerned with long-time dynamics of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. We also establish the existence of a global attractor, and we study the fractal dimension and exponential attractors.

Citation: M. L. Santos, Mirelson M. Freitas. Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1869-1890. doi: 10.3934/cpaa.2019087
References:
[1]

M. S. AlvesJ. E. Mu noz 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. Mu noz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat J. Solids Struct., 24 (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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.

[5]

V. Barbu, Analysis and Control of Nonlinear Infnite-Dimensional Systems, vol. 190, Mathematics in Science and Engineering, Academic Press Inc, Boston, 1993.,

[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]

Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689–722.

[9]

I. ChueshovM. 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.

[10]

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

[11]

B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete and Continuous Dynamical Systems, 37 (2017), 4729-4751.  doi: 10.3934/dcds.2017203.

[12]

B. Feng, T. F. Ma, R. N. Monteiro and C. A. Raposo, Dynamics of laminated timoshenko beams, J Dyn Diff Equat, (2017). doi: 10.1007/s10884-017-9604-4.

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical surveys and monographs, American Mathematival Society, Providence, RI, 1988.

[14]

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

[15]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.

[16]

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.

[17]

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

[18]

P. PeiM. A. Rammaha and D. Toundykov, Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations, Nonlinear Analysis, 105 (2014), 62-85.  doi: 10.1016/j.na.2014.03.024.

[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.

[20]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SIAM, Philadelphia, PA, 1995.

show all references

References:
[1]

M. S. AlvesJ. E. Mu noz 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. Mu noz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Internat J. Solids Struct., 24 (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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.

[5]

V. Barbu, Analysis and Control of Nonlinear Infnite-Dimensional Systems, vol. 190, Mathematics in Science and Engineering, Academic Press Inc, Boston, 1993.,

[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]

Diffusion in mixtures of elastic materials, Int. J. Eng. Sci., 7 (1969), 689–722.

[9]

I. ChueshovM. 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.

[10]

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

[11]

B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete and Continuous Dynamical Systems, 37 (2017), 4729-4751.  doi: 10.3934/dcds.2017203.

[12]

B. Feng, T. F. Ma, R. N. Monteiro and C. A. Raposo, Dynamics of laminated timoshenko beams, J Dyn Diff Equat, (2017). doi: 10.1007/s10884-017-9604-4.

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical surveys and monographs, American Mathematival Society, Providence, RI, 1988.

[14]

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

[15]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.

[16]

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.

[17]

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

[18]

P. PeiM. A. Rammaha and D. Toundykov, Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations, Nonlinear Analysis, 105 (2014), 62-85.  doi: 10.1016/j.na.2014.03.024.

[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.

[20]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SIAM, Philadelphia, PA, 1995.

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