On a nonlinear wave equation associated with the boundary conditions involving convolution

The paper deals with the initial boundary value problem for the linear wave equation {u(tt) - partial derivative/partial derivative x(mu(x, t)u(x)) + lambda u(t) + F(u) = 0, 0 < x < 1, 0 < t < T, mu(0, t)u(x)(0, t) = g(0)(t) + integral(t)(0) k(0)(t - s)u(0, s)ds, (1) -mu(1, t)u(x)(1, t) = g(1)(t) + integral(t)(0) k(1)(t - s)u(1, s)ds, u(x, 0) = u(0)(x), u(t)(x, 0) = u(1)(x), where F, mu, g(0), g(1), k(0), k(1), u(0), u(1) are given functions and lambda is a given constant. The paper consists of four main parts. In Part 1, under conditions (u(0), u(1), g(0), g(1), k(0), k(1)) is an element of H-1 x L-2 x (H-1(0, T))(2) x (W-1,W-1(0, T))(2), mu is an element of W-1(Q(r)), mu(t), is an element of L-1(0, T; L-infinity), mu(x, t) >= mu(0) > 0, a.e. (x, t) is an element of Q(T); the function F continuous. integral(z)(0) F(s)ds >= -C(1)z(2) - C-1', for all z is an element of R, with C-1, C-1' > 0 are given constants and some other conditions, we prove that, the problem (1) has a unique weak solution u. The proof is based on the Faedo-Galerkin method associated with the weak compact method. In Part 2 we prove that the unique solution u belongs to H-2 (Q(T)) boolean AND L-infinity(0, T; H-2) boolean AND C-0(0, T; H-1) boolean AND C-1(0, T; L-2), 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), if we assume (u(0), u(1)) is an element of H-2 x H-1, F is an element of C-1 (R) and some other conditions. In Part 3, with F is an element of CN+1 (R), N >= 2, we obtain an asymptotic expansion of the solution u of the problem (1) up to order N + 1 in a small parameter lambda. Finally, in Part 4, we prove that the solution u of this problem is stable with respect to the data (lambda, mu, g(0),g(1), k(0), k(1)). (C) 2008 Elsevier Ltd. All rights reserved.