Global Solution In Non-homogeneous Boundary Conditions For Non-linear Beam Equation With Memory Term

At present,with the development of practical problems and the advancement of mathernatics itself,the research on infinite-dimensional dynamic systems has become one of the important subjects of dynamic systems.In this paper, we consider a equation under non-linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non-linear bearing at x=L. By adding only one damping mechanism at x=L,we prove the existence of a global solution.The details will go as follows,(1)Firstly,the current study situation and research methods about general infinite-dimensional dynamical system in solid mechanics, and points out the research back-ground of this article, also gave an analysis of the correlation equation.(2)Secondly,on the basis of Woinowsky-Krieger beam equation at the hinge model, by adding memory and nonlinear function M, we set up a more general equation about viscous elastic beam.(3)For the vibration of the beam model we studied the equation with the initial conditions u(x,0)=u0(x), ut(x,0)=u1(x) and the nonlinear boundary conditions u(0,t)=ux(0,t)=0 under the above conditions of initial boundary value problems.Through transformthe equation, structure the approximate solution of the equa-tion, prior estimate and combined with some inequalities technique.spatial of Soblev space,convergence of the equation and take the steps to limit,To prove that the ex-istence,uniqueness and constant dependence of global solution. Also studied the expo-nential decay by defining the energy of system.