Existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions

Consider the initial-boundary value problem for the nonlinear wave equation u(tt) - partial derivative/partial derivative x (mu(x, t)u(x)) + f (u, u(t)) = F(x, t), 0 < x < 1, 0 < t < T, mu(0, t)u(x)(0,t) = P(t), -mu(1, t)u(x) (1, t) = K-1\u(1, t)\(p1-2)u(1, t) + \u(t)(1, t)\(q1-2)u(t)(1, t), u (x, 0) = u(0) (x), u(t) (x, 0) = u(1)(x) where p(1), q(1) >= 2, K-1 >= 0 are given constants and mu, u(0), u(1), f, F are given functions, and the unknown function u(x, t) and the unknown boundary value P (t) satisfy the following nonlinear integral equation t P(t) = g(t) + K-0\u(0, t)\(p0-2)u(0, t) + \u(t)(0, t)\(q0-2)u(t)(0, t) - integral(t)(0) k(t - s)u(0, s)ds, (2) where p(0), q(0) >= 2, K-0 >= 0 are given constants and g, k are given functions . In this paper, we consider three main parts. In Part 1, under the conditions (u(0), u(1), F, g, k) is an element of H-1 x L-2 x L-1 (0, T; L-2) x L-q0'(0, T) x L-1(0, T), mu is an element of C-0((Q(T)) over bar), mu(x, t) >= mu(0) > 0, mu(t) is an element of L-1(0, T; L-infinity), mu(t) (x, t) <= 0, a.e. (x, t) is an element of Q(T); K-0, K-1 >= 0; p(0), q(0), p(1), q(1)>= 2, q(o') = q0/q(0-1), the function f supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others, we prove that the problem (1) and (2) has a weak solution (u, P). If, in addition, k is an element of W-1,W-1 (0, T), p(0), p(1) is an element of [2] boolean OR [3, +infinity) and some other conditions, then the solution is unique. The proof is based on the Faedo-Galerkin method and the weak compact method associated with a monotone operator. For the case of q(0) = q(1) = 2; p(0), p(1) >= 2, in Part 2 we prove that the unique solution (u, P) belongs to (L-infinity(0, T; H-2) boolean AND C-0(0, T; H-1) boolean AND C-1(0, T; L-2)) x H-1(0, T), with u(t) is an element of L-infinity(0, T; H-1), u(tt) is an element of L-infinity(0, T; L-2), u(0,.), u(1,.) is an element of H-2(0, T), if we assume (u(0), u(1)) is an element of H-2 x H-1, f is an element of C-1(R-2) and some other conditions. Finally, in Part 3, with q(0) = q(1) = 2; p(0), p(1) >= N + 1, f is an element of CN+1 (R-2), N >= 2, we obtain an asymptotic expansion of the solution (u, P) of the problem (1) and (2) up to order N + 1 in two small parameters K-0, K-1. (c) 2006 Elsevier Ltd. All rights reserved.