TITLE:
Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations
AUTHORS:
Guoguang Lin, Lujiao Yang
KEYWORDS:
Generalized Kirchhoff Equation, The Existence and Uniqueness of Solution, A Family of the Global Attractor, Dimension Estimation
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.11 No.4,
April
25,
2021
ABSTRACT: In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress termM (s)in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0kis obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroupS (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.