On the nonlinear wave equation u(tt)-B(t,parallel to u parallel to(2),parallel to u(x)parallel to(2),parallel to u(t)parallel to(2) )u(xx)=f(x, t,u, u(x), u(t),parallel to u parallel to(2),parallel to u(x)parallel to(2),parallel to u(t)parallel to(2)

摘要

In this paper we consider the following nonlinear wave equation:u(n) - B(t, parallel touparallel to(2), parallel tou(x)parallel to(2), parallel toutparallel to(2))u(xx) = f(x, t, u, u(x), u(t), parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2)),x is an element of (0, 1), 0 < t < T, (1)u(x)(0, t) - h(0)u(0, t) = u(x)(1, t) + h(1)u(1, t) = 0, (2)u(x, 0) - (u) over tilde (0)(x), u(t)(x, 0) = (u) over tilde (i)(x), (3)where h(0) > 0, h(1) greater than or equal to 0 are given constants and B, f, (u) over tilde (0), (u) over tilde (1) are given functions. In Eq. (1), the nonlinear terms B(t, parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2)), f(x, t, u, u(x), u(t), parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2)) depending on the integrals parallel touparallel to(2) = integral(Omega) u(x, t)(2) dx, parallel tou(x)parallel to(2) = integral(0)(1)u(x)(x, t)(2) dx and parallel toutparallel to(2) = integral(Omega)u(t)(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 CN+1(R-+(4)), B greater than or equal to b(0) greater than or equal to 0, B-1 is an element of C-N (R-+(4)), B-1 greater than or equal to 0, f is an element of CN+1([0, 1] x R+ x R-3 x R-+(3)) and f(1) is an element of C-N([0, 1] x R+ x R-3 x R-+(3)) we obtain from the following equation u(n) - [B(t, parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2))+ epsilonB(1) (t, parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2))]u(xx) = f(x, t, u, u(x), u(t), parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2)) + epsilonf(1)(x, t, u, u(x), u(t), parallel touparallel to(2), parallel tou(x)parallel to(2), parallel tou(t)parallel to(2)) associated to (2), (3) a weak solution u(epsilon)(x, t) having an asymptotic expansion of order N + 1 in epsilon, for epsilon sufficiently small. (C) 2004 Elsevier Ltd. All rights reserved.