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 utt - uxx + ƒ(u, ut) = 0, x ∈ Ω = (0, 1), 0 < t < T, ux(0, t) = P(t), u(1, t) = 0, u(x, 0) = u0(x), ut(x, 0) = u1(x), where u0, u1, ƒ are given functions, the unknown function u(x, t) and the un-known boundary value P(t) satisfy the nonlinear integral equation P(t) = g(t) + H(u(0, t)) - ∫t0 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, H and K. For the proof, we use the Galerkin method.