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Eden, A., Foias, C., Nicolaenko, B. and Teman, R. (1991) Inertial Sets for Dissipative Evolution Equations Part I: Construction and Applications. University of Minnesota, Twin Cities.

has been cited by the following article:

• TITLE: The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

  AUTHORS: Guoguang Lin, Chunmeng Zhou

  KEYWORDS: Family of Exponential Attractors, Lipschitz Continuous, Squeezing Property, Stochastic Dynamic System, Family Random Attractors

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.9 No.12, December 27, 2021

  ABSTRACT: To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets Bk. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system; thereby we obtain a family of random attractors.