A singularly perturbed Cauchy problem

Abstract

Singular perturbations are considered for the initial value problem for second order hyperbolic equations with a small positive parameter multiplying the second order time derivative term. Thus, in contrast to recent work of de Jager and Geel, the reduced equation is of the same order as the original equation but of a different type. Asymptotic expansions are constructed and shown to be uniformly asymptotically valid on sets bounded in the time direction. The proof uses energy estimates which require some delicacy due to the dependence of the characteristics on the small parameter. The problem includes that of a vibrating string in a highly viscous medium when appropriate scaling is made. The proper initial conditions differ from those treated by Zlamal and the methods employed here are different as well.