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

In this paper we consider the following nonlinear wave equation ut t − B t, ux 2 uxx = f (x, t,u,ux,ut ), x ∈ Ω = (0, 1), 0 < t 0, B1 ∈ C1(R2 +), B1 0, f ∈ C2(Ω × [0,∞) × R3) and f1 ∈ C1(Ω × [0,∞) ×R3) we obtain from the following equation ut t − B t, ux 2 + εB1 t, ux 2 uxx = f (x, t,u,ux,ut ) + εf1(x, t,u,ux,ut ) associated to (2), (3) a weak solution uε(x, t ) having an asymptotic expansion of order 2 in ε, for ε sufficiently small.