On the nonlinear wave equation U-tt-B(t, parallel to U-x parallel to(2))U-xx = f(x, t, U, U-x, U-t) associated with the mixed homogeneous conditions

摘要

In this paper we consider the following nonlinear wave equation (1) u(tt) - B (t, parallel tou(x)parallel to(2))u(xx) = f (x, t, u, u(x), u(t)), x is an element of Omega = (0, 1), 0 < t < T, (2) u(x)(0, t) - h(0)u (0, t) = 0, u(x)(1, t) + h(1)u(1, t) = 0, (3) U(x, 0) = (u) over tilde0(x), u(t)(x, 0) = (u) over tilde (1)(x), where h(0), h(1) are given nonnegative constants and B, f, (u) over tilde (0), (u) over tilde (1) are given functions. In Eq. (1) the coefficient B(t, parallel tou(x)parallel to(2)) containing an integral parallel tou(x)parallel to(2) = integral(0)(1)/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-2(R-2), B greater than or equal to b(0) > 0, B-1 is an element of C-1(R-+(2)), B-1 greater than or equal to 0, f is an element of C-2((&UOmega;) over barx [0, infinity] x R-3) and f(1) is an element of C-1 ((&UOmega;) over bar x [0, infinity) x R-3) we obtain from the following equation u(tt) - (B (t, parallel tou(x)parallel to(2)) + epsilonB(1)(t, parallel tou(x)parallel to(2)))u(xx) = f(x, t, u, u(x), u(t)) + epsilonf(1) (x, t, u, u(x), u(t)) associated to (2), (3) a weak solution mu(epsilon) (x, t) having an asymptotic expansion of order 2 in epsilon, for epsilon sufficiently small. (C) 2002 Elsevier Science (USA). All rights reserved. [References: 12]