TITLE:
The Family of Global Attractors for a Generalized Kirchhoff Equations
AUTHORS:
Guoguang Lin, Xiaomei Liu
KEYWORDS:
Kirchhoff Equation, Galerkin’s Method, Family of Global Attractor, Dimension Estimation
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.11 No.7,
July
20,
2021
ABSTRACT: This paper deals with the existence and uniqueness of solutions of generalized Kirchhoff equations and the family of global attractors for the equation and its dimension estimation. First, the stress term of Kirchhoff equation is properly assumed. When certain conditions are met between the order m and the degree p of Banach space , the existence and uniqueness of the solution of equation are obtained by a prior estimation and Galerkin’s method; Then, the bounded absorption set is obtained by prior estimation, and it is proved that the solution semigroup generated by the equation has a family of global attractors in phase space by using Rellich-Kondrachov compact embedding theorem. Further, the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Fréchet differentiable on ; Finally, the upper bound of Hausdorff dimension and Fractal dimension of is estimated, and the Hausdorff dimension and Fractal dimension are finite.