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

Abstract

We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t})=0,quad xin Omega =(0,1),; 0<t<T, cr u_{x}(0,t)=P(t),quad u(1,t)=0, cr u(x,0)=u_0(x),quad u_{t}(x,0)=u_1(x), }$$ 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.

Keywords

• Galerkin method
• integrodifferential equations
• Schauder fixed point theorem
• weak solutions
• stability of the solutions