June  2012, 32(6): 2079-2088. doi: 10.3934/dcds.2012.32.2079

A minimal approach to the theory of global attractors

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 101447, Russian Federation

2. 

Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133, Italy, Italy

Received  March 2011 Revised  June 2011 Published  February 2012

For a semigroup $S(t):X\to X$ acting on a metric space $(X,d)$, we give a notion of global attractor based only on the minimality with respect to the attraction property. Such an attractor is shown to be invariant whenever $S(t)$ is asymptotically closed. As a byproduct, we generalize earlier results on the existence of global attractors in the classical sense.
Citation: Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.

[3]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[4]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Masson, Paris, 1991.

[5]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Elsevier/North-Holland, Amsterdam, (2008), 103-200.

[6]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[7]

V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equation with nonlinear damping, Adv. Math. Sci. Appl., 17 (2007), 225-237.

[8]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[9]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.

[3]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.

[4]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Masson, Paris, 1991.

[5]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Elsevier/North-Holland, Amsterdam, (2008), 103-200.

[6]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[7]

V. Pata and S. Zelik, Attractors and their regularity for 2-D wave equation with nonlinear damping, Adv. Math. Sci. Appl., 17 (2007), 225-237.

[8]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[9]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.

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