On the nonlinear wave equation utt −B(t, u 2, ux 2)uxx = f (x, t,u,ux,ut , u 2, ux 2) associated with the mixed homogeneous conditions

In this paper we consider the following nonlinear wave equation: (1) utt −B(t, u 2, ux 2)uxx = f (x, t,u,ux,ut , u 2, ux 2), x ∈ (0, 1), 0 < t 0, h1 0 are given constants and B, f , ˜u0, ˜u1 are given functions. In Eq. (1), the nonlinear terms B(t, u 2, ux 2), f (x, t,u,ux,ut , u 2, ux 2) depend on the integrals u 2 = Ω |u(x, t)|2 dx and ux 2 = 1 0 |ux(x, t)|2 dx. In this paper I associate with problem (1)–(3) a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument. In case of B ∈ CN+1(R3 +), B b0 > 0, B1 ∈ CN(R3 +), B1 0, f ∈ CN+1([0, 1]×R+ ×R3 ×R2 +) and f1 ∈ CN([0, 1]×R+ ×R3 ×R2 +) we obtain for the following equation utt − [B(t, u 2, ux 2) + εB1(t, u 2, ux 2)]uxx = f (x, t,u,ux,ut , u 2, ux 2) + εf1(x, t,u,ux,ut , u 2, ux 2) associated to (2), (3) a weak solution uε(x, t) having an asymptotic expansion of order N + 1 in ε, for ε sufficiently small.  2004 Elsevier Inc. All rights reserved