| At present, with the development of practical problems and some other sciences , especially ,with the advancement of mathematics itself on the other, the research on infinite-dimensional dynamics systems has become one of the important subjects of dynamic systems. The research of the limited-dimensional dynamic systems of the dynamics systems there has been 30 years' history at least, and obtain many important results. But the problem of the dynamics systems does not be limited by the circumstance of the limited-dimensional far and far. we have discovered a large quantity nonlinear evolution equation with soliton, For example kdv equation, nonlinear Schrodinger equation, Zakharov equation etc, under the function of dissipation evolve from the soliton to chaos. in addition, some dissipation partial differential equations, such as: reaction-diffusion equation, Navier-stokes equation, Kuramoto-Sivashinskey equation etc, has similar phenomen -on. The research on infinite-dimensional dynamics systems is already necessity certainly at line, it have the some new important characteristic. This thesis presents some research on nonlinear damping Sine-Gordon beam equation and give the proof of existence and uniqueness under The details will go as follows.Firstly, the current study situation and research methods about general infinite-dimensional dynamic systems are summarized and commented. Particularly, the threshod of the infinite-dimensional dynamic systems—the existence, uniqueness of solutions has been discussed, especially, the problem of initial boundary for Sine-Gordon equation is summarized.Secondly, on the basis of Roger-Temam's Sine-Gordon model, we study a more general nonlinear damping Sine-Gordon beam equation by the aid of Galerkin method, under boundary conditionsand under initial conditions:we prove the existence, uniqueness of a local weak solution , and the constant dependence of solution about the initial value.
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