Totally dissipative dynamical processesand their uniform global attractors

Vladimir V. Chepyzhov; Monica Conti; Vittorino Pata

doi:10.3934/cpaa.2014.13.1989

Article Contents

2014, Volume 13, Issue 5: 1989-2004. Doi: 10.3934/cpaa.2014.13.1989

This issue Previous Article Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law Next Article The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction

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: September 2015

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 34D45, 37L30, 47H20.

Citation:

Related Papers

Cited by

References

[1]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
[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.
[3]	V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors, Russian J. Math. Phys., 1 (1993), 165-190.
[4]	V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
[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.
[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.
[7]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, 2002.
[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.
[9]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.
[10]	A. Haraux, Systèmes dynamiques dissipatifs et applications, Coll. RMA no.17, Masson, Paris, 1991.
[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.
[12]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.doi: 10.1007/978-1-4757-5037-9.
[13]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.doi: 10.1007/978-1-4612-0645-3.