Long Time Behavior Of A Class Of Non-Linear High-low-Order Coupled Kirc-Hhoff Equations

This paper studies the long-term behavior of a class of nonlinear high-low-order coupled Kirchhoff equations.It mainly discusses the three aspects of the global attrac-tor,exponential attractor and inertial manifold of the equations.Under the appropriate assumptions,the existence and uniqueness of the solution of the equations is obtained on E-1=H-0~2(?)×H-0~2m(?)×H-0~1(?)×H-0~m(?)through the prior estimation and Galerkin method,and then we obtain the semigroup of the equations,and further verify the ex-istence of the global attractor.Based on the study of the global attractor of the equa-tions,it is verified that the solution semigroup of the equations are squeezed,and satisfy the Lipschitz continuity.Therefore,the equations have the exponential attractor,and the exponential attractor has a finite fractal dimension.Finally,by transforming the second-order equations into first-order evolution equations,constructing the correspond-ing graphs,exploring that the spectral gap condition is true,we get the inertial manifold of the equations.