On the nonlinear wave equation Ut t − B(t, Ux 2)Uxx = f (x, t,U,Ux,Ut , Ux 2) associated with the mixed nonhomogeneous conditions

In this paper we consider the following nonlinear wave equation ut t −B t, ux 2 uxx = f x, t,u,ux,ut , ux 2 , x∈ Ω = (0, 1), 0 < t 0, B1 ∈ C2(R2+ ), B1 0, f ∈ C3([0, 1]×R+ ×R3 ×R+) and f1 ∈ C2([0, 1]×R+ ×R3 ×R+) we obtain from the equation ut t −[B(t, ux 2)+εB1(t , ux 2)]uxx = f (x, t,u,ux,ut , ux 2)+εf1(x, t,u, ux,ut , ux 2) associated to (2), (3) a weak solution uε(x, t ) having an asymptotic expansion of order 3 in ε, for ε sufficiently small.  2004 Elsevier Inc. All rights reserved