Initial-Boundary Value Problem For Beam Equation Disspative Term Under The Impact Of Nonlinear Force

In the modern application of mathematics,elastic beam equations of the scholars has been one of the important issues, including the beam equation with dissipation highly valued by people.This article discusses a class of nonlinear forces and internal stress i n the joint effect of the beam with dissipation equation with:Where R,α,γis an arbitrary constant,andβ∈C1且α"≥β'(s)≥α'(α'α"is positive constant), F(u,u)=f1(u)+f2(u), where f1(u) is nonlinear force term, f2(u) is viscous damping, which are b ounded in the bounded-set,Ωis abounded convex domain with smooth b oundary (?)Ωin Rn, andΔis Laplace operator,▽is a gradient operator and u,u stand respectively a second order and a first order partial derivatives, what is more,‖·‖is the usual sense norm of the L2(Ω) and unknown functi on u(x,t) is displacement, u0(x),u1(x) are known functions in t=0.Specific contents are as follows:First, this article introduces the domestic and international research on the status of the Nonlinear Beam Equation.Second, the article gives some important concepts and lemmas and some symbols.Third, the existence and uniqueness of weak solution of the equations (1)-(3) are proved by Galerkin method.Fourth, the strong solutionof the equations (1)-(3)are proved by Galerkin method.Fifth, further evidence of the initial boundary value problem of the global solution of the continuous dependence on initial conditions.