On the nonlinear wave equation U-tt-B(t, vertical bar vertical bar U-x vertical bar vertical bar(2)) U-xx = f(x, t, U, U-x, U-t, vertical bar vertical bar U-x vertical bar vertical bar(2)) associated with the mixed nonhomogeneous conditions

摘要

In this paper we consider the following nonlinear wave equation u(tt) - B(t, parallel tou(x) parallel to(2))u = f (x, t, u, u(x), u(t) parallel tou(x parallel to)(2)), x is an element of Omega = (0, 1), 0 < t < T, (1)u(x)(0, t) - h(0)u(0, t) = g(0)(t), u(1, t) = g(1) (t), (2)u(x, 0) = u(0)(X), ut(x, 0) = u(1) (X), (3)where B, f, g(0), g(1),g(0), u(0),u(l) are given functions. In Eq. (1), the nonlinear terms B(t, parallel toux parallel to(2)), f (x, t, u, u(x), u(t), parallel tou(x) parallel to(2)) depending on an integral parallel tou(x) parallel to(2) = f (1)(0) u(x) (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 standard compactness argument. In case of B is an element of C-3(R-+(2)), B greater than or equal to b(0) > 0, B-1 is an element of C-2(R-+(2)), B-1 greater than or equal to 0, f is an element of C-3([0.1] x R+ x R-3 x R+) and f(1) is an element of C-2([O, 1] x R+ x R-3 x R+) we obtain from the equation u(tt) - [B(t, parallel to u(x) parallel to(2)) + epsilonB(1) (t, parallel to u(x) parallel to(2))]u(xx) = f (x, t, u, u(x), u(t), parallel tou(x) parallel to(2)) +epsilonf(1) (x, t, u, u(x), u(t), parallel tou(x) parallel to(2)) associated to (2), (3) a weak solution u(epsilon) (x, t) having an asymptotic expansion of order 3 in epsilon, for epsilon sufficiently small. (C) 2004 Elsevier Inc. All rights reserved.