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

Totally dissipative dynamical processes and their uniform global attractors

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.
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, (1992).

[2]

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

[3]

V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors,, \emph{Russian J. Math. Phys.}, 1 (1993), 165.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, \emph{J. Math. Pures Appl.}, 73 (1994), 279.

[5]

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

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of periodic processes and estimates of their dimensions,, \emph{Math. Notes}, 57 (1995), 127. doi: 10.1007/BF02309145.

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).

[8]

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

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).

[10]

A. Haraux, Systèmes dynamiques dissipatifs et applications,, Coll. RMA no.17, (1991).

[11]

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

[12]

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

[13]

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

show all references

References:
[1]

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

[2]

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

[3]

V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors,, \emph{Russian J. Math. Phys.}, 1 (1993), 165.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, \emph{J. Math. Pures Appl.}, 73 (1994), 279.

[5]

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

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of periodic processes and estimates of their dimensions,, \emph{Math. Notes}, 57 (1995), 127. doi: 10.1007/BF02309145.

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).

[8]

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

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).

[10]

A. Haraux, Systèmes dynamiques dissipatifs et applications,, Coll. RMA no.17, (1991).

[11]

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

[12]

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

[13]

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

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