The Existence And Uniqueness Of Global Solution For A Class Of The Nonlinear Beam Equation With Damping

The partial differential equation is the important part of modern mathema-tics, and its theories are applied to the practical problems in physics, chemistry, biology, mechanics and so on. The nonlinear partial differential equation is part-icularly valued, and the existence and uniqueness of its solution, one of the hot spots is paid more attention to.Due to the complexity of the nonlinear partial differential equation, it is dif-ficult or even impossible to obtain the analytical solutions for the most nonlinear partial differential equations. We can only acquire the numerical solutions.So people take advantages of the equation of own characteristics to estimate the properties of solutions.In order to provide theoretical basis for the practical application, it is necessary to study the existence and uniqueness of its solution, and it is also essential to research the global solution in the nonlinear beam equations. In this paper, we will discuss a class of the nonlinear beam equations with damping, and prove the existence and uniqueness of the global solution for this kind equation under certain conditions.In this paper, we will study a kind of the nonlinear damping beam equation by means of the Galerkin method in the generalized function space under the initial conditions u(x,0)=u0(x) u(x,0)=u1(x)(2) and the boundary conditions u(0,t)=u(1,t)=u(2)(0,t)=u(2)(1,t)=0(3) where B,f,u0,u1, are given functions. In the equation(1),nonlinear terms depend on integrationThis article will be divided into the following four-part study:1. We will make some introduction on the development and research of nonlinear partial differential equation.2. This paper will give some important definition and lemmas, simulta-neity explain some marks.3. We will prove the existence and uniqueness of weak solutions by Galerkin method.4. We will prove the existence and uniqueness of strong solutions by Galerkin method.