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.