Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary
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Date
2007-03-20
Authors
Nguyen, Thanh Long
Truong, Le Xuan
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
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.
Description
Keywords
Nonlinear wave equation, Linear integral equation, Existence and uniqueness, Asymptotic expansion
Citation
Long, N. T., & Truong, L. X. (2007). Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary. <i>Electronic Journal of Differential Equations, 2007</i>(48), pp. 1-19.
Rights
Attribution 4.0 International