The Long Time Behavior For A Class Of High-order Nonlinear Kirchhoff Beam Equations With Strong Damping

The Kirchhoff equation is the most basic and important type of nonlinear partial differential equations.It can be used to describe the lateral vibration of the elastic rope and the population density problem in ecology.This paper studies the long time dynamic behavior of a class of high order Kirchhoff-type beam equations with nonlinear strong damping terms.Under appropriate assumptions,we use prior estimation and Galerkin method to prove that the existence and uniqueness of the global solution of the equation.Then,we construct the bounded absorption set of the solution semigroup S（t）,and prove that the solution semigroup S（t）is uniformly bounded and completely continuous in Ek.Thus,the global attractor family of the equation is obtained,and we further obtain the finite Hausdorff dimension and Fractal dimension of the global attractor family.Next,we change the external force term g（x）to the random term q（x）W,and use the Ornstein-Uhlenbeck process to transform the equation into a noise-free random process with random variables as parameters.Finally,the existence of inertial manifold family is proved by using the operator semigroup theory and Hadamard graph transformation method under certain spectral interval conditions.