A nonlinear wave equation with a nonlinear integral equation involving the boundary value

We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t})=0,quad xin Omega =(0,1),; 0 Keywords Galerkin method --- integrodifferential equations --- Schauder fixed point theorem --- weak solutions --- stability of the solutions