Asymptotics of resonances in a thermoelastic model with light local mass perturbations

Abstract

The limit behaviour of a linear one-dimensional thermoelastic system with local mass perturbations is studied. The mass density is supposed to be nearly homogeneous everywhere except in an ε \varepsilon -vicinity of a given point, where it is of order ε − m \varepsilon ^{-m} , with m ∈ R m \in \mathbb {R} . The resonance vibrations of the string are investigated as ε → 0 \varepsilon \to 0 . An important ingredient of the analysis is the construction of an operator in a space of higher regularity such that its spectrum coincides with that of the classical operator in linearised thermoelasticity, with a correspondence of generalised eigenspaces. The convergence of eigenvalues and eigenprojectors is established along with error bounds for two classes of relatively light mass perturbations, m > 1 m>1 and m = 1 m=1 , which exhibit contrasting limit behaviour.