Existence, uniqueness and asymptotic behaviour for a nonlinear beam equation

The use of perturbation methods for fourth order PDE's has not yet been examined extensively. Usually approximating power series are applied, which are truncated to one or two modes. Very little - or nothing - is said about the relation between this approximation and the exact solution. In this paper the following equation will be discussed: w(tt) + w(xxxx) + epsilon (u(pi, t) - u(O,t) + integral(o)(pi)w(x)(2)dx)w(xx) = epsilon g(x,t,w, w(t)). This equation can be regarded as a model describing wind-induced oscillations of flexible structures like elastic beams, where the small term on the right hand side of the equation represents the windforce acting on the structure. Existence and uniqueness for solutions Of these problems will be discussed, as well as finding approximations using a multiple time scales method. Finally the asymptotic validity of these approximations will be considered.