Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary

Abstract

We study the initial-boundary value problem for the nonlinear wave equation utt - ∂/∂x (μ(x, t)ux) + K|u|p-2 u + λ|ut|q-2 ut = ƒ(x, t), u(0, t) = 0 -μ(1, t)ux (1, t) = Q(t), u(x, 0) = u0(x), ut(x, 0) = u1(x), where p ≥ 2, q ≥ 2, K, λ are given constants and u0, u1, ƒ, μ are given functions. The unknown function u(x, t) and the unknown boundary value Q(t) satisfy the linear integral equation. Q(t) = K1(t)u(1, t) + λ1(t)ut (1, t) - g(t) - ∫t0 k(t - s)u(1, s)ds, where K1, λ1, g, k are given functions satisfying some properties stated in the next section. This paper consists of two main sections. First, we prove the existence and uniqueness for the solutions in a suitable function space. Then, for the case K1(t) = K1 ≥ 0, we find the asymptotic expansion in K, λ, K1 of the solutions, up to order N + 1.