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doi: 10.3934/dcdsb.2021036

Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system

Mustapha Yebdri

Laboratory of dynamical systems, Department of Mathematics, Faculty of sciences, University of Tlemcen, Tlemcen, BP.119, 13000 Algeria

Received  April 2020 Revised  December 2020 Published  January 2021

At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.

For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.

In this paper we give a positive aftermath by proving the existence of pullback $ \mathcal{D} $-attractor. The key point is to find a bounded family of pullback $ \mathcal{D} $-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback $ w $-$ \mathcal{D} $-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.

Keywords: Infinite dimenisional dynamical systems, partial differential equations with delay, asymptotic behavior, pullback attractor, $ \omega- $limit set.
Mathematics Subject Classification: Primary: 35B41, 37L30.
Citation: Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021036
References:
[1]

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

[2]

J. Barrow-Green, Poincaré and the three-body problem, History of Mathematics, vol. 11., Amer. Math.Soc., Providence, RI, 1997. doi: 10.1090/hmath/011.  Google Scholar

[3]

P. Bergé, Y. Pomeau and C. Vidal, L'ordre Dans le Chaos, , Hermann, Paris, 1984. Google Scholar

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J. E. Bilotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088.  doi: 10.1090/S0002-9904-1971-12879-3.  Google Scholar

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E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. D ifferential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.  Google Scholar

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M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J.Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

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H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.  Google Scholar

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I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[11]

H. Haraga and M. Yebdri, Pullback attractors for class of semilinear nonclassical diffusion equation with delay, Applied Mathematics and Nonlinear Sciences, 3 (2018), 127-150.  doi: 10.21042/AMNS.2018.1.00010.  Google Scholar

[12]

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

[13]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, J. Applied Mathematics and Computation, 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[14]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris 1969.  Google Scholar

[15]

X. Liu and Y. Wang, Pullback attractors for nonautonomous 2D-Navier-Stokes models with variable elays, J. Abstract and Appl. Anal., (2013), Art. ID 425031, 10 pp. doi: 10.1155/2013/425031.  Google Scholar

[16] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, (2nd edition), Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

C. The Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, J. Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[19]

Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, J. Discrete and Continuous Dynamical Systems, 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.  Google Scholar

[20]

J. Wu, Theory and Applications of Partiali Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

show all references

References:
[1]

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

[2]

J. Barrow-Green, Poincaré and the three-body problem, History of Mathematics, vol. 11., Amer. Math.Soc., Providence, RI, 1997. doi: 10.1090/hmath/011.  Google Scholar

[3]

P. Bergé, Y. Pomeau and C. Vidal, L'ordre Dans le Chaos, , Hermann, Paris, 1984. Google Scholar

[4]

J. E. Bilotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088.  doi: 10.1090/S0002-9904-1971-12879-3.  Google Scholar

[5]

E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. D ifferential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.  Google Scholar

[6]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J.Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[7]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.  Google Scholar

[8]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[11]

H. Haraga and M. Yebdri, Pullback attractors for class of semilinear nonclassical diffusion equation with delay, Applied Mathematics and Nonlinear Sciences, 3 (2018), 127-150.  doi: 10.21042/AMNS.2018.1.00010.  Google Scholar

[12]

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

[13]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, J. Applied Mathematics and Computation, 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[14]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris 1969.  Google Scholar

[15]

X. Liu and Y. Wang, Pullback attractors for nonautonomous 2D-Navier-Stokes models with variable elays, J. Abstract and Appl. Anal., (2013), Art. ID 425031, 10 pp. doi: 10.1155/2013/425031.  Google Scholar

[16] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[17]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, (2nd edition), Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

C. The Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, J. Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[19]

Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, J. Discrete and Continuous Dynamical Systems, 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.  Google Scholar

[20]

J. Wu, Theory and Applications of Partiali Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

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