Dynamic Behavior Of A Class Of Kirchhoff Equations With High-order Strong Dissipation

In this thesis,we mainly study the dynamic behavior of a class of high-order Kirchhoff equation with strong dissipation with four parts.Before this,the rigid term in the equation is assumed according to the truncation method,and the critical indices of the rigid term p and the nonlinear source term ρ are given.In the first part,a priori estimate of the solution of the problem is made.The evidence that there exists solution of the initial boundary value problem is given via constructing the approximate solution and taking the limit with Galerkin’s method.That is,there exists a solution semigroup of the equation.Then the difference method is used to prove the uniqueness of the solution.In the second part,the family of the global attractor of the initial boundary value problem is proved via constructing the bounded absorption set.Then the problem is linearized,the solution semigroup is proved that is Frechet differentiable in the space Ek.Then,the finite dimensional estimate of Hausdorff dimension and Fractal dimension of the family of global attractor can be obtained.In the third part,the problem is transformed into a first-order development equation,and it is proved that the eigenvalues of the operator satisfy the spectral interval condition through the case discussion.By using the Hadamard’s graph transformation theory,it is proved that the initial boundary value problem has a family of inertial manifold and its dimension estimation is obtained.In the fourth part,the random equation is transformed via Ornstein-Uhlenbeck process δ(θtω),and the priori estimate of the solution of the transformed equation is made to obtain the bounded random absorption set B0k via the property of O-U process |δ(θtω)|.Finally,by proving the existence of asymptotically compact random absorption closed set Jk(ω)in stochastic dynamical systems,we obtain that the stochastic equations have the family of global attractor.