A semilinear wave equation associated with a nonlinear integral equation

Abstract

The paper deals with the initial-boundary value problem for the semilinear wave equation (Equation presented) where u0, u1, f are given functions, the unknown function u(x, t) and the unknown boundary value P(t) satisfy the following nonlinear integral equation (Equation presented) where g, H, k are given functions. We prove the existence and uniqueness of weak solutions to the problem, and discuss the stability of the solution (u, P) with respect to the functions g, H and k. In the proof, the Galerkin method is employed. © 2013 Warsaw University. All rights reserved.