Long time approximations for solutions of wave equations via standing waves from quasimodes

Abstract

A quasimode for a positive, symmetric and compact operator on a Hilbert space could be defined as a pair ( u , λ ), where u is a function approaching a certain linear combination of eigenfunctions associated with the eigenvalues of the operator in a “small interval” [ λ − r , λ + r ] . Its value in describing asymptotics for low and high frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been made clear recently in many papers. In this paper, considering second order evolution problems, we provide estimates for the time t in which standing waves of the type e i λ t u approach their solutions u ( t ) when the initial data deal with quasimodes of the associated operators. We establish a general abstract framework and we extended it to the case where operators and spaces depend on the small parameter ε: now λ and u can depend on ε and also perform the estimates for t. We apply the results to vibrating systems with concentrated masses.