A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

Abstract

In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.