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

Babych, Natalia; Zimmer, Johannes

doi:10.1090/S0033-569X-09-01112-8

Authors: Natalia Babych and Johannes Zimmer
Journal: Quart. Appl. Math. 67 (2009), 311-326
MSC (2000): Primary 35P15; Secondary 34E10, 74F05
DOI: https://doi.org/10.1090/S0033-569X-09-01112-8
Published electronically: March 20, 2009
MathSciNet review: 2514637
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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 $\varepsilon$-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 $\varepsilon \to 0$. 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, $m<1$ and $m=1$, which exhibit contrasting limit behaviour.

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