Linear Recursive Schemes and Asymptotic Expansion Associated with the Kirchoff–Carrier Operator

In this paper we consider a nonlinear wave equation with the Kirchoff–Carrier operator, utt − b0 + B ∇u 2 u = f x t u ux ut x ∈ = 0 1 0 < t < T u 0 t = u 1 t = 0 u x 0 = ˜ u0 x ut x 0 = ˜ u1 x 1 2 3 where b0 > 0 is a given constant and B f are the given functions. In Eq. (1) the function B ∇u 2 depends on the integral ∇u 2 = ∇u x t 2 dx. In this paper we associate with problem (1)–(3) a linear recursive scheme for which the existence of a local and unique solution is proved by using the standard compactness argument. If B ∈ C2 R + B1 ∈ C1 R + f ∈ C2 × 0 ∞ × R3 , and f1 ∈ C1 × 0 ∞ × R3 then an asymptotic expansion of order 2 in ε is obtained with a right-hand side of the form f x t u ux ut + εf1 x t u ux ut , and B stands for B + εB1, for ε sufficiently small.