Long-Time Behavior of Second-Order Abstract Equations

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


The main aim in this chapter is to present general methods that can be used in the study of the long-time behavior of dynamical systems generated by damped second-order abstract equations. To accomplish this we use abstract results of Chapter 7 that describe long-time dynamics of more general dynamical systems. Particular assumptions imposed on the data of the problem (damping and sources) are motivated by applications to nonlinear plate equations, with particular emphasis on von Karman evolutions. In fact, in Chapter 9 we specialize these results to von Karman models with internal nonlinear damping and both subcritical and critical sources. However, abstract hypotheses formulated are tailored to accomodate larger classes of nonlinear dynamic plate equations. We refer to [75] for a more general approach covering a larger class of nonlinear second order evolution equations and to [103, 123, 124, 156] for the the case of the corresponding abstract semilinear problems with linear damping.


Strong Solution Global Attractor Stabilizability Estimate Energy Inequality Nonconservative Force 


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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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