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

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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

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