Existence Of Local Solutions To A Class Of Nonlinear Wave Equations

Currently, nonlinear science has become an important frontier of scientific research. In recent decades, many researchers from home and abroad do a lot of research and get a lot of results in terms of solving the nonlinear wave equations. Now scientists have developed a number of methods for solving nonlinear wave equations.This paper studied the existence of local solution nonlinear wave equations with bound-ary condition and nonlinear wave equations with variable coefficients. In other words, we study and improve some methods of solving nonlinear wave equations with boundary condi-tion. And the result is effective for nonlinear wave equation with variable coefficients. The issues not only have very important theory value but also have broad application prospects. Therefore, the study of these issues has attracted considerable attention, and fruits from research into it emerge continuously. However, some of methods only can be used to linear wave equations. In this paper, we get the existence of local solution of a class of nonlinear wave equations with boundary condition. And by using this method we obtain the existence of local solution of a class of nonlinear wave equations with variable coefficients.This article includes three chapters.The first chapter is an introduction. It is a brief introduction about existence of local solution of nonlinear wave equations and mainly results of this article.The second chapter we introduct the first class of nonlinear wave equations with bound-ary condition studied in this paper where (0, L) is a bounded open set in R and p> 1, b> 0.In this chapter, we studied the existence of local solutions of nonlinear wave equations with initial boundary value condition. Similarly, we get the existence of local solution of a class of nonlinear wave equations by using Galerkin approximation method. And the process is divided into three steps. The first step is the Galerkin approximation. The second step is a priori estimate and the third step is take the limit process. Finally, we get the existence of local solutions of nonlinear wave equations with constant coefficient.The third chapter we introduct the second class of nonlinear wave equations with vari-able coefficient studied in this paper, where (0, L) is a bounded open set in R, and p> 1, b≥0, a(x)∈C1(0, L).In this chapter, we studied the existence of local solution of nonlinear wave equations with variable coefficient. We get the existence of local solution of a class of nonlinear wave equations by using Galerkin approximation method. And the process is divided into three steps. The first step is the Galerkin approximation. The second step is a priori estimate and the third step is take the limit process. Finally, we get the existence of local solution of nonlinear wave equations with variable coefficients.