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Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation

Published online by Cambridge University Press:  30 January 2014

Alexandre N. Carvalho ,
Jan W. Cholewa  and
Tomasz Dłotko
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 protected])
Jan W. Cholewa
Affiliation:
Institute of Mathematics, Silesian University, 40-007 Katowice, Poland ([email protected]; [email protected])
Tomasz Dłotko
Affiliation:
Institute of Mathematics, Silesian University, 40-007 Katowice, Poland ([email protected]; [email protected])
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Abstract

We consider a family of bounded dissipative asymptotically compact semigroups depending on a parameter, and study the continuity properties of the corresponding family of its global attractors. We exploit the idea of the uniform exponential attraction property to discuss the continuity properties of the family of attractors and estimate the rate of convergence of the approximating attractors to the limit one. Showing a range of applications of an abstract framework, we focus much of our attention on a perturbed damped wave equation. In this latter case our results involve nonlinearities with critical exponents, for which the continuity of the family of attractors is concluded, including the rate of convergence and the regularity of the limit attractor. This complements the results in the literature.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 1 , February 2014 , pp. 13 - 51
Copyright
Copyright © Royal Society of Edinburgh 2014 

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