Long Time Behavior Of A Class Of Nonlinear Kirchhoff Equations

This paper studies the initial boundary value problem for a class of nonlinear coupled Kirchhoff equations utt-M(??u?2+??v?2)?u-??ut+?(1+|u|2)pu=f1(x),vtt-M(??v?2+??v?2)?v-??vt+?(1+|v|2)pv=f2(x).Firstly,some a priori estimates are obtained under appropriate assumptions.In addition,the existence and uniqueness of the solutions are obtained by a priori estim-ation and applying the Galerkin method.Constructing a bounded absorption set and solving The compactness of the semigroup to obtain the global attractor.Based on the existence of the global attractor,by linearizing the equations to solve the semigroup's differentiability and volume element attenuation are obtained the global attractor has a finite Hausdorff dimension,We use the Hadamard graph transformation method to obtain the equivalent form of the original equation,and let its linear differential opera-tor be a positive definite operator under the graph.Then,the existence of inertial manifold of the equation is discussed by using the spectral gap condition.