On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

Abstract

In this paper, we consider the following nonlinear wave equation (1){(ut t - frac(∂, ∂ x) (μ (u) ux) = f (x, t, u, ux, ut), 0 < x < 1, 0 < t < T,; ux (0, t) = g (t), u (1, t) = 0,; u (x, 0) = over(u, ̃)0 (x), ut (x, 0) = over(u, ̃)1 (x),) where over(u, ̃)0, over(u, ̃)1, μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo-Galerkin method and the weak compact method. In the case of μ ∈ CN + 2 (R), μ1 ∈ CN + 1 (R), μ (z) ≥ μ0 > 0, μ1 (z) ≥ 0, for all z ∈ R, and g ∈ C3 (R+), f ∈ CN + 1 ([0, 1] × R+ × R3), f1 ∈ CN ([0, 1] × R+ × R3), a weak solution uε1, ε2 (x, t) having an asymptotic expansion of order N + 1 in two small parameters ε1, ε2 is established for the following equation associated to (1)2,3: (2)ut t - frac(∂, ∂ x) ([μ (u) + ε1 μ1 (u)] ux) = f (x, t, u, ux, ut) + ε2 f1 (x, t, u, ux, ut) . © 2009 Elsevier Ltd. All rights reserved.