| We consider a second-order semilinear evolution equation in a real Hilbert space, with Cauchy data and a small parameter (epsilon):;u(0) = u(,0), (epsilon)u'(0) = u(,1). (I).;Many nonlinear damped vibration problems of mechanics can be formulated in the abstract setting of (E) (I). This problem is singularly perturbed since there is a reduction of order when (epsilon) = 0 and the reduced equation.;U'(t)+AU(T) = F(U(t)).;(epsilon)('2)u''(t)+u'(t)+Au(t) = F(u(t)) (E).;together with the single condition U(0) = u(,0), forms a well-posed problem. This behavior suggests the existence of an initial layer. Our main result is the proof of the uniform validity of an N-term asymptotic expansion of u with initial layer corrections on an arbitrary time interval {0,T}. We thereby succeed in extending to an abstract setting recent results of Hsiao and Weinacht dealing with the semilinear partial differential equation.;(epsilon)('2)u(,tt)+u(,t)-u(,xx) = f(u).;This abstract analysis is subsequently used to obtain estimates for a continuous-time numerical scheme for a class of singularly perturbed second-order hyperbolic pde's in (//R)('n). Convergence of the scheme by finite element methods is obtained. |
