A Family Of Attractors For A Class Nonlinear High-order Kirchhoff Equations

In this paper,we study the well posedness,global attractor family and its dimension estimation,stochastic attractor family of solutions for high-order Kirchhoff type equations with nonlinear source term and strong damping term.Under appropriate assumptions,we use prior estimation and Galerkin method to prove that there is a unique global solution(u,v)? L?([0,+?);Ek)(k=0,1,2,…,m).Then we construct the bounded absorption set of solution semigroup S(t),and prove that solution semigroup S(t)is uniformly bounded and completely continuous in Ek,thus the global attractor family of the equation is obtained,furthermore,the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.In addition,the external force term f(x)is changed to the random term q(x)W,and the original high-order Kirchhoff equation is simplified to the first-order evolution equation by using the ornstin-Uhlenbeck process and Ito equation.Secondly,it is proved that the first-order evolution equation has a bounded absorption set in D(Ek),there is a compact absorption set K(?)for stochastic dynamical system {S(t,?),t?0} when k=1,2,…,m.At last,a stochastic dynamical system {S(t,?),t? 0} with random attractor family(?)k is obtained.