Long-time Behavior Of A Class Of High-order Nonlinear Kirchhoff Equations

In this paper,we discuss the well-posedness and the long-time dynamic behavior of the initial-boundary value problem for higher-order Kirchhoff equations.By adopting appropriate assumptions for M(s)and g(ut)of the equation and according to a prior estimate and the Galerkin’s method,it is proved that the equations have unique and unique solutions.Semigroup S(t)of solutions of a defined equation,it is easy to know that S(t)has a family of global attractors.After the equation is linearized,it is easy to prove the Frechet differentiability of S(t).Finally,two finite dimensions of the family of global attractors are obtained,which are Hausdorff and Fractal.By exploring the family of global attractors,it is easy to know that a solution semigroup S(t)satisfies not only Lipschitz property,but also squeezing property.Thus,it is verified that there are exponential attractors.Then,the spectral interval condition is established when N is sufficiently large.Finally,the existence of the family of inertial manifolds is proved.