Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations
- 1 Department of Mathematics, Yunnan University, Kunming, China
- 2 Department of Mathematics, Yunnan University, Kunming, China
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 term M ( 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 B 0 k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S ( t ) generated by the equation has a family of the global attractor A k in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on E k . Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor A k was obtained.
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