The Application Of Topological Methods In Nonlinear Wave Equations

It is a core of the theory of research field of differential equation to study multiplicity results for differential equations under boundary conditon. Also ,it is one of the key subjects of the research contents of this field.It has backgrounds of deep physics and mechanics to utilize the theories of topological degree, variational reduction method and critical point principle to investigate solvability and multiplicity results of differential equations under boundary condition. Solving the problem needs topological and geometrical properties of classical space . At the same time, the settlement of the problem drives a lot of new production and development of tool in nonlinear analysis , and has shown a multi-disciplinary research field that blends each other. Through a large number of mathematicians' efforts, the theory of this field has already formed a kind of typical treatment method for partial differential equations.This text utilizes these methods mainly, study solvability of the several differential equations and multiplicity of solutions on the basis of forefathers.1. The first chapter introduces several symbols, definitions and some results about wave equation and beam equaiton ,which have been obtained by using the theories of topological degree , variational reduction method and critical point principle .2. In the second chapter, we mainly investigate a nonlinear wave equation under periodic condition on the variable t . First,we use the variational method to reduce the problem from an infinite dimensional one to a finite dimensional one . Then,we investigate the properties of the map Φ and reveal a relation between multipicity of solution and source terms h(x,t) in equation when h(x,t) is generated by the eigenfunction φ₀₁ and φ₀₂.3. In the third chapter, we mainly investigate solutins of the beam equation . Fist, we know the solutions coincide with the critical points of I|^<sub>b{v) by variational method. Then, we investigate some properties of I|^<sub>b(v). By critical points theory, we may prove I|^<sub>b(v) has at least three critical points, so the beam equation also has at least three solutions.