Attractors for Evolutionary Equations

Part of the Springer Monographs in Mathematics book series (SMM)


This chapter provides a survey of quantitative theory pertinent to long-time behavior of infinite-dimensional dissipative systems. The results are presented in a convenient form for applications to the material presented in subsequent chapters. For other possible approaches to the topic we refer to the monographs [17, 61, 134, 139, 172, 259, 273].

The main focus is on questions such as existence of global attractors, and their structure, dimensionality, and smoothness. In this context, gradient systems play a prominent role as an important subclass of more general dissipative systems. With the aim of unifying specific criteria that lead to the desired properties of attractors (mentioned above), we single out a class of “quasi -stable” systems that enjoy the so-called stabilizability–observability inequality. This inequality, although often difficult to establish, once it is proved provides a string of consequences that describe various properties of attractors.


Banach Space Global Attractor Inertial Manifold Exponential Attractor Dissipative Dynamical System 


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© Springer Science+Business Media 2010

Authors and Affiliations

  1. 1.Department of Mechanics & MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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