Geodesic
A nonlinear wave equation with a nonlinear integral equation involving the boundary value
Electronic journal of differential equations, Tome 2004 (2004)
We consider the initial-boundary value problem for the nonlinear wave equation

$\displaylines{ u_{tt}-u_{xx}+f(u,u_{t})=0,\quad x\in \Omega =(0,1),\; 0, \cr u_{x}(0,t)=P(t),\quad u(1,t)=0, \cr u(x,0)=u_0(x),\quad u_{t}(x,0)=u_1(x), }$

where

$ 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.
Classification : 35B30, 35L70, 35Q72
Keywords: Galerkin method, integrodifferential equations, Schauder fixed point theorem, weak solutions, stability of the solutions 
@article{EJDE_2004__2004__a92,
     author = {Nguyen,  Thanh Long and Bui,  Tien Dung},
     title = {A nonlinear wave equation with a nonlinear integral equation involving the boundary value},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1073.35175},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a92/}
}
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%A Bui,  Tien Dung
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Nguyen,  Thanh Long; Bui,  Tien Dung. A nonlinear wave equation with a nonlinear integral equation involving the boundary value. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a92/