January  2022, 27(1): 167-198. doi: 10.3934/dcdsb.2021036

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

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 2022 Early access  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.

Citation: Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036
References:
[1]

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

[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.

[3]

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

[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.

[5]

E. M. BonottoM. C. BortolanT. 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.

[6]

M. C. BortolanA. 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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

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

[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.

[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.

[20]

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

show all references

References:
[1]

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

[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.

[3]

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

[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.

[5]

E. M. BonottoM. C. BortolanT. 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.

[6]

M. C. BortolanA. 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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

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

[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.

[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.

[20]

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

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