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September  2014, 13(5): 1989-2004. doi: 10.3934/cpaa.2014.13.1989

Totally dissipative dynamical processes and their uniform global attractors

Vladimir V. Chepyzhov 1, , Monica Conti 2, and  Vittorino Pata 2,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 101447
2. 

Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received  February 2013 Revised  April 2013 Published  June 2014

We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$. Such an attractor is uniform with respect to $\sigma\in\Sigma$, as well as with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established for totally dissipative processes without any continuity assumption. When the process satisfies some additional (but rather mild) continuity-like hypotheses, a characterization of the attractor is given.
Keywords: absorbing and attracting sets, Dynamical processes, uniform global attractors..
Mathematics Subject Classification: 34D45, 37L30, 47H2.
Citation: Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1989-2004. doi: 10.3934/cpaa.2014.13.1989
References:
[1]

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

[2]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dynam. Systems, 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors, Russian J. Math. Phys., 1 (1993), 165-190.  Google Scholar

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Periodic processes and non-autonomous evolution equations with time-periodic terms, Topol. Methods Nonlinear Anal., 4 (1994), 1-17.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of periodic processes and estimates of their dimensions, Math. Notes, 57 (1995), 127-140. doi: 10.1007/BF02309145.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, 2002.  Google Scholar

[8]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping, Glasg. Math. J., 48 (2006), 419-430. doi: 10.1017/S0017089506003156.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[10]

A. Haraux, Systèmes dynamiques dissipatifs et applications, Coll. RMA no.17, Masson, Paris, 1991.  Google Scholar

[11]

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.  Google Scholar

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G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[13]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  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]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dynam. Systems, 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors, Russian J. Math. Phys., 1 (1993), 165-190.  Google Scholar

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Periodic processes and non-autonomous evolution equations with time-periodic terms, Topol. Methods Nonlinear Anal., 4 (1994), 1-17.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of periodic processes and estimates of their dimensions, Math. Notes, 57 (1995), 127-140. doi: 10.1007/BF02309145.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, 2002.  Google Scholar

[8]

S. Gatti and V. Pata, A one-dimensional wave equation with nonlinear damping, Glasg. Math. J., 48 (2006), 419-430. doi: 10.1017/S0017089506003156.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[10]

A. Haraux, Systèmes dynamiques dissipatifs et applications, Coll. RMA no.17, Masson, Paris, 1991.  Google Scholar

[11]

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.  Google Scholar

[12]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[13]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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