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Exponential global attractors for semigroups in metric spaces with applications to differential equations

Published online by Cambridge University Press:  15 March 2011

ALEXANDRE N. CARVALHO
Affiliation:
Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil (email: andcarva@icmc.usp.br)
JAN W. CHOLEWA
Affiliation:
Institute of Mathematics, Silesian University, 40-007, Katowice, Poland (email: jcholewa@ux2.math.us.edu.pl)

Abstract

In this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Arrieta, J., Carvalho, A. N. and Lozada-Cruz, G.. Dynamics in dumbbell domains II. The limiting problem. J. Differential Equations 247 (2009), 174202.CrossRefGoogle Scholar
[2]Babin, A. V. and Vishik, M. I.. Attractors of Evolution Equations. North-Holland, Amsterdam, 1992.Google Scholar
[3]Bruschi, S. M., Carvalho, A. N., Cholewa, J. W. and Dlotko, T.. Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations. J. Dynam. Differential Equations 18 (2006), 767814.CrossRefGoogle Scholar
[4]Carvalho, A. N.. Contracting sets and dissipation. Proc. Roy. Soc. Edinburgh A 125 (1995), 13051329.CrossRefGoogle Scholar
[5]Carvalho, A. N. and Langa, J. A.. The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces. J. Differential Equations 233 (2007), 622653.CrossRefGoogle Scholar
[6]Carvalho, A. N. and Langa, J. A.. An extension of the concept of gradient semigroups which is stable under perturbation. J. Differential Equations 246 (2009), 26462668.CrossRefGoogle Scholar
[7]Cholewa, J. W. and Dlotko, T.. Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar
[8]Eden, A., Foias, C., Nicolaenko, B. and Temam, R.. Exponential Attractors for Dissipative Evolution Equations. John Wiley & Sons, Ltd., Chichester, 1994.Google Scholar
[9]Hale, J. K.. Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence, RI, 1988.Google Scholar
[10]Ladyzenskaya, O.. Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge, 1991.CrossRefGoogle Scholar
[11]Raugel, G.. Global Attractors in Partial Differential Equations (Handbook of Dynamical Systems, 2). North-Holland, Amsterdam, 2002, pp. 885982.Google Scholar
[12]Sell, G. R. and You, Y.. Dynamics of Evolutionary Equations. Springer, New York, 2002.CrossRefGoogle Scholar
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