Abstract
This chapter explores the asymptotic behavior for a strongly damped nonlinear wave equation. The boundedness of orbits in one space implies boundedness of orbits in other spaces. Invariant sets in one space are automatically invariant sets in many spaces, which implies smoothness properties of invariant sets. Point dissipative and compact dissipative are equivalent in many spaces and imply bounded dissipative in spaces of smoother functions. The chapter reviews the existence of a very smooth maximal compact invariant set under a very weak dissipative assumption, along with its strong stability and attractivity properties in several spaces. It discusses the existence, uniqueness, and the variation of constants formula, along with certain compactness properties of the orbits for an equation. It also discusses boundedness of orbits and describes invariant sets.
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