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 e (0, 1), 0 < t < T, (2) ux(0, t) - h0u(0, t) = ux(1, t) + h1u(1, t) = 0, (3) u(x,0)=∥0(x),ut(x,0)=∥1(x), where ho > 0, h1 ≥ 0 are given constants and B, f,?0,?1 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 = ∫10 |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 e CN+1(R3+), B ≥ by > 0, B1 ∈ CN(R3+), B1 ≥ 0, f ∈ CN+1([0, 1] × R+ × R3 × R2+) and f1 ∈ CN([0, 1] × R+ × R3 x 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 e, for e sufficiently small.