Long Time Behavior Of A Class Of High-order Kirchhoff-type Equations

This paper discusses the long-term behavior of a class of nonlinear higher-order Kirchhoff equations.Under the appropriate assumptions of Kirchhoff stress terms and second-order nonlinear source terms,on the one hand,using a priori estimation,Galerkin's method to construct an approximate solution uh=uh(t)=(?)gjh(t)wj,taking the limit and obtained the existence of the only global solution(u,v)?L?([0,+?);Ek).(k=1,2,…m)of the equation.The total continuity of the bounded absorption set B0k and the solution semigroup S(t):Ek?Ek is constructed from the prior estimation,and the existence of the global attractor family is proved.Then,we linearize the equation Pt+??P+(?)(?)P=0,prove the Frechet differentiability of the solution semigroup,and obtain the finite dimensional estimates of Hausdorff dimension and fractal dimension of the global attractor family;On the other hand,the external force term f(x)of the Kirchhoff equation is replaced with white noise q(x)W.The Ornstein-Uhlenbeck process is used to transform the stochastic equation,and the solution of the equation is estimated to obtain a bounded random absorption set B0k(?)?D(Ek).Then,using the compact embedding theorem Ek(?)E0,it is obtained that the stochastic dynamic system {S(t,w),t?0} is progressively compact at t=0,Pa.e.???,thereby proving the family of random attractors.