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

摘要

In this paper we consider the following nonlinear wave equation:(1) u(tt) - B(t, parallel to u parallel to(2), parallel to u(x)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)), x is an element of(0,1), 0 < t < T, (2) u(x)(0,t) - h(0)u(0,t) = u(x)(1,t) + h(1)u(1,t) = 0, (3) u(x,0) = (u) over tilde (x), u(t)(x,0) = (u) over tilde (x),where h(0) > 0, h(1) >= 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 to u parallel to(2)' parallel to u(x)parallel to(2)), f(x, t, u, u(x), u(t), parallel to u parallel to(2) , parallel to u(x)parallel to(2)) depend on the integrals 0 parallel to u parallel to(2) integral(Omega)vertical bar u(x,t)vertical bar(2) dx and parallel to u(x),parallel to(2) = integral(0)(1) vertical bar u(x)(x,t)vertical bar(2)dx. In this paper I associate with problem (l)-(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-3(+)), B >= b(0) > 0, B-1 is an element of C-N(R-3(+)), B-1 >= 0, f integral is an element of C-N+1([0, 1] xR(+) x R-3 xR(+)(2)) and f(1) is an element of C-N([0, 1] x R+ x R-3 xR(2)(+)) we obtain for the following equation u(tt) - [B(t,parallel to u parallel to(2), parallel to u(x)parallel to(2)) + epsilon B-1(t, parallel to u parallel to(2), parallel to u(x)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)) + epsilon f(1)(x, t, u, u(x), u(t), parallel to u parallel to(2), parallel to u(x)parallel to(2)) associated to (2), (3) a weak solution u, (x, t) having an asymptotic expansion of order N + I in epsilon, for epsilon sufficiently small. (c) 2004 Elsevier Inc. All rights reserved.