The Global Dynamics Of Elastic Beam Equation With Nonlinear Boundary Conditions

At present,with the development of practical problems and some other sciences on the one hand, and with the advancement of mathematics itself on the other,the research on infinite-dimensional dynamic systems has become one of the important subjects of dynamic systems.This thesis presents some research on nonlinear elastic infinite-dimensional dynamic systems of solid mechanics and we give the proof of existence and the asymptotic behavior under certain initial boundary conditions.The details will go as follows:Firstly,the current study situation and research methods about general infinite-dimensional dynamic systems are summarized and commented.Particularly,the threshold of the infinite-dimensional dynamic systems—the existence,uniqueness of solutions has been discussed.Moreover, the exponential decay of the energy is summarized.Secondly,on the basis of Woinowsky-Krieger bar vibra-tion model with axial force effect,simultaneously considering the viscous effect of material;geometrical nonlinear-ity; physical nonlinearity;periodical axial loads;we set up a more general nonlinear equation about viscous elastic beam.For the nonlinear viscolastic beam system:by the aid of Galerkin method,under nonlinear boundary conditions(a):and under initial conditions:we prove the existence,uniqueness of a global solution and the constant dependence of solution about the initial value. Moreover,the proof of exponential decay of the energy is studied as well. At the same time,Galerkin method is also applied under the upper initial conditions and nonlinear boundary conditions(b):the proof of existence,uniqueness and constant dependence for initial boundary value of global solution are illustrated.