Some Studies On One Dimensional Nonlinear Wave Equation

In this paper,firstly we study the Cauchy problem of one-dimension fully nonlinear wave equations with null condition,by transforming the one-dimension fully nonlinear wave equation to a system of one-dimension quasilinear wave equations,and using the result in the quasilinear case,we show the global existence of classical solution with small initial data.Then we consider the initial-boundary value problem on R+ ×R+for some one dimensional systems of quasilinear wave equations with null conditions.Firstly we prove that for homogeneous Dirichlet boundary values and sufficiently small initial data,classical solutions always globally exist.Then we show that the global solution will scatter,i.e.,it will converge to some solution of one dimensional linear wave equations as time tends to infinity,in the energy sense.Finally we prove the following rigidity result:if the scattering data,vanish,then the global solution will also vanish identically.