Study On Posedness Of Solutions To The Nonlinear Beam Equations With Different Damping

Partial differential equations are divided into linear partial differential equations and nonlinear partial differential equations. Many problems in the field of science and engineering can be attributed to study of the nonlinear partial differential equations. It is difficult to acquire exact solutions for most differential equations, especially for nonlinear partial differential equations. Thus, study of approximate solutions to the differential equations has important theory significance. The research on the existence and uniqueness of solutions of differential equations is a foundation for approximate calculation.Our main work includes that summarize and comment the development and actuality on partial differential equation or equations and carry out some studies on the posedness of the solutions to the two kinds of nonlinear partial differential equations by means of the Sobolev space.Firstly, we prove the existence and uniqueness of weak solution and strong solution to the nonlinear beam equation with weak damping subject to the boundary conditions and the initial conditionsSecondly, we show the existence and uniqueness of the weak solution to the nonlinear beam equation with structural damping u uuuquMl udxNluudxuf+subject to the boundary conditions and the initial conditions Whereμ,η,q are real numbers and . The functions which are defined in [ 0,∞) are nonnegative and continuously differentiable.Thirdly, we prove the solution of the following nonlinear partial differential equation subject to the initial conditions (2) and the boundary conditions (3), its solution blows up in finite time. And there f (u) is a nonlinear function.