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

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 $\varepsilon$-vicinity of a given point, where it is of order $\varepsilon^{-m}$, with $m \in \mathbb{R}$. The resonance vibrations of the string are investigated as $\varepsilon\to0$. 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, $m<1$ and $m=1$, which exhibit contrasting limit behaviour.