On a nonlinear wave equation with a nonlocal boundary condition

Abstract

Consider the initial-boundary value problem for the nonlinear wave equation utt − µ(t)uxx + K|u| p−2u + λ|ut| q−2ut = F(x, t), 0 < x < 1, 0 < t < T, µ(t)ux(0, t) = K0u(0, t) + Rt 0 k (t − s) u (0, s) ds + g(t), −µ(t)ux(1, t) = K1u(1, t) + λ1|ut(1, t)| α−2ut(1, t), u(x, 0) = ue0(x), ut(x, 0) = ue1(x), where p, q, α ≥ 2; K0, K1, K ≥ 0; λ, λ1 > 0 are given constants and µ, F, g, k, ue0, ue1, are given functions. First, the existence and uniqueness of a weak solution are proved by using the Galerkin method. Next, with α = 2, we obtain an asymptotic expansion of the solution up to order N in two small parameters λ, λ1 with error p λ2 + λ 2 1 N+ 1 2 .