The Long Time Behavior Of Solutions For A Class Of Generalized Beam-Kir-Chhoff Type Equations

In this paper,global attractor families,inertial manifolds and pullback attractor families of generalized Beam-Kirchhoff type equations are studied.The global attractor family reflects the compact invariance of the solution in space,the inertial manifold connects the infinite dimension to the finite dimension space,and the invariance and attractivity of the pullback attractor family are valid for all trajectories at each starting point of the solution.In this paper,Galerkin finite element method is used to prove the existence and uniqueness of the solution of the equation under the appropriate assumption of Kirchhoff term.The solution semigroup is defined,and the compact bounded absorption set is obtained by priori estimation.The existence of desorption initiator for infinite dimensional dynamical systems is discussed,which provides a condition for the existence of global attractor for solutions.On this basis,it is proved that the solution semigroup is uniformly bounded and completely continuous.Further,it is obtained that the solution of the equation has a compact global attractor family.In addition,the Frechet differentiability of the solution semigroup is verified by linearization of the equation,and the Hausdorff dimension and Fractal dimension of the global attractor family are estimated.By constructing the graph norm corresponding to the equation,the existence of inertial manifolds of the equation is obtained under certain spectral interval conditions.In addition,the delay term is added to the original equation,and the pull-back attractor family of the solution of the equation is studied by using the method of constructing limit in the environment of non-autonomous system.These properties provide a theoretical basis for solving the models of fluid mechanics,mathematical physics and bridge vibration,as well as for solving the problems in real life.For example,the existence of geochemical landscape attractors is of great significance to the selection of geochemical sampling density.