Dynamic Behavior Of A Class Of Higher-order Kirchhoff Equations

In this thesis,we study the dynamic behavior of higher-order Kirchhoff equation.First part,the stress M（‖▽^mu‖_p^p）of Kirchhoff equation is properly assumed.When certain conditions are met between the order m and the degree p of Banach space L^p（Ω）,the uniform prior estimation of time and Galerkin’s method are used,In order to obtain the existence and uniqueness of the solution of this equation,prior estimation is needed.Then,the bounded absorption set B_0k is obtained by prior estimation,and it is proved that the solution semigroup S（t）generated by the equation has a family of global attractors_kA in phase space E_k=(H^2m+k（Ω）∩H ₀¹（Ω）)×H ₀^k（Ω）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 S（t）is Frechet differentiable on E_k;If we want to get that Hausdorff dimension and Fractal dimension are finite,we estimate the upper bound of these two types of dimension of A_k.The second part,firstly,in order to obtain the existence of exponential attractors for initial-boundary value problems,we should first prove the Lipschitz property and squeezing property of nonlinear semigroups related to initial-boundary value problems.By extending the space E₀ to E_k,a family of the exponential attrator is obtained.Next,use the graph exchange method.we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.