Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 103, pp. 1-21.

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

Authors: Thanh Long Nguyen (Vietnam National Univ. HoChiMinh, Vietnam)
         Tien Dung Bui (Univ. of Architecture of HoChiMinh City, Vietnam)
 
Abstract: 
 We consider the initial-boundary value problem for the 
 nonlinear wave equation 
 $$\displaylines{
 u_{tt}-u_{xx}+f(u,u_{t})=0,\quad x\in \Omega =(0,1),\; 0<t<T, \cr
 u_{x}(0,t)=P(t),\quad u(1,t)=0, \cr
 u(x,0)=u_0(x),\quad u_{t}(x,0)=u_1(x),
 }$$
 where $u_0, u_1, f$ are given functions, the unknown function 
 $u(x,t)$  and the unknown boundary value $P(t)$ satisfy the
 nonlinear integral equation 
 $$
 P(t)=g(t)+H(u(0,t))-\int_0^t K(t-s,u(0,s))ds,
 $$
 where $g$, $K$, $H$ are given functions. We prove the existence 
 and uniqueness of weak solutions to this problem, and discuss 
 the stability of the solution with respect to the functions 
 $g$, $H$ and $K$. For the proof, we use the Galerkin method.
 
Submitted January 14, 2004. Published September 3, 2004.
Math Subject Classifications: 35B30, 35L70, 35Q72.
Key Words: Galerkin method; integrodifferential equations;
Schauder fixed point theorem; weak solutions;
stability of the solutions