Global Attractors And Approximate Inertial Manifolds For A Class Of Kirchhoff Equations

In this paper,we first study the global solutions of the initial boundary value prob-lems for a class of Kirchhoff equations with nonlinear strong damping,and then the global attractors and approximate inertial manifolds of the infinite dimensional dynamical sys-tems corresponding to the equations are also proved.The specific equations are as follows utt-?1?ut???ut?p-1 ut???u?q-1u-?(||?u||2)?u= f(x)(x,t)??×R+,u(x,0)= uo(x);ut(x,0)= u1(x)x ??,Where? is a bounded domain in RN with smooth boundary(?)?,?(|| ?U||2)is a nonlinear function,f(x)?L2(?)is an external term and ?1,?,? are positive constants.Firstly,we prove the existence and uniqueness of the global solution of the above problems by a uniform priori estimate and Galerkin method.Furthermore,the global attractor is obtained by using the theory of semigroups of operators.In order to construct an approximate inertial manifold for the problems considered in this paper,we use the properties of the analytic semigroup generated by the linear differential operator in the phase space.Finally,we construct an approximate inertial manifold for the problems considered in this paper.