Electron. J. Diff. Eqns., Vol. 2004(2004), No. 103, pp. 1-21.

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

Thanh Long Nguyen, Tien Dung Bui

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 $g$, $K$, and $H$. 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

Show me the PDF file (323K), TEX file, and other files for this article.

Thanh Long Nguyen
Department of Mathematics and Computer Science
University of Natural Science
Vietnam National University HoChiMinh City
227 Nguyen Van Cu Str., Dist.5, HoChiMinh City, Vietnam
email: longnt@hcmc.netnam.vn
Tien Dung Bui
Department of Mathematics
University of Architecture of HoChiMinh City
196 Pasteur Str., Dist. 3, HoChiMinh City, Vietnam

Return to the EJDE web page