Initial-Boundary Value Problems For A Kind Of Nonlinear System Of Partial Differential Equations

Partial differential equation is an important bridge between mathematics theory and actual application. The research on partial differential equation based on problem in physics or mechanics and so on, is not only a most important content of traditional applied mathematics but also a significant portion of modern mathematics. With the research progressing, to some problem that can be approximately solved formerly with linear partial differential equation, we must consider the effect of nonlinear factor now. So, the study on partial differential equation focuses on nonlinear partial differential equation. The qualitative analysis to nonlinear partial differential equation is very difficult, and it is a big trouble to deal it in a common way, just as linear equations. When study them, we are apt to combine relevant actual model tightly.In recent years, the qualitative study on nonlinear qualitative partial differential equation mainly focuses on the existence of partial solutions (gobal solutions don't exist perhaps), the existence of global solutions, regularity and the estimation on the energy decrease.In this paper , we present some results concerning the following nonlinear system of partial differential equationscü+γ(v|¨)+δv⁽⁴⁾-β₀v⁽²⁾=f₂(t). (2)The above system is a mathematical model which describes coupled flexural and torsional oscillations of an open cross-section beam.we consider the problem of finding u and v solutions of the system(1)-(2),verifying the initial conditionsu(x,0)=u₀(x),(u|·)(x,0)=u₁(x), (3)v(x,0)=v₀(x),(v|·)(x,0)=v₁(x), (4)and the boundary conditionsu(0,t)=u(1,t)=u⁽²⁾(0,t)=u⁽²⁾(1,t)=0, (5)v(0,t)=v(1,t)=v⁽²⁾(0,t)=v⁽²⁾(1,t)=0. (6)The particular content is following.1.We made simple sum-up and comment on the developing and actuality of atudy on partial differential equation and comment on the developing and actuality of study on partial differential equation and equations relevant with this paper.2.We give some important definitions and lemmas,3.With Galerkin method, through three steps: approximate solution, prophetic estimate, constringency, we imparted the prove to the existence and uniqueness of weak solutions of nonlinear system of beam equations (1)-(6).4.Proved the existence and uniqueness of the strong solution.5.Proved the existence and uniqueness of the classics solution.