In this paper we consider a nonlinear wave equation with the Kirchoff-Carrier operator,where b 0 > 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 ∈ C 2 R + B 1 ∈ C 1 R + f ∈ C 2 × 0 ∞ × R 3 , and f 1 ∈ C 1 × 0 ∞ × R 3 then an asymptotic expansion of order 2 in ε is obtained with a right-hand side of the form f x t u u x u t + εf 1 x t u u x u t , and B stands for B + εB 1 , for ε sufficiently small.  2002 Elsevier Science (USA)

We treat an initial boundary value problem for a nonlinear wave equation u tt − u xx + K|u| α u + λ|u t | β u t = f (x,t) in the domain 0 < x < 1, 0 < t < T. The boundary condition at the boundary point x = 0 of the domain for a solution u involves a time convolution term of the boundary value of u at x = 0, whereas the boundary condition at the other boundary point is of the form u x (1,t) + K 1 u(1,t) + λ 1 u t (1,t) = 0 with K 1 and λ 1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α = β = 0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N + 1 in two small parameters K, λ.

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