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 ε-vicinity of a given point, where it is of order ε−m, with m ∈ R. The resonance vibrations of the string are investigated as ε→ 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. 1 Problem statement We consider resonance vibrations of a finite string modelled in the framework of linearised, onedimensional thermo-elasticity. The evolution of the displacement u = u(x, t) and the relative temperature θ = θ(x, t) is governed by the system of differential equations [9, 8, 5] ρε(x) ∂u ∂t2 − ∂ ∂x ( α(x) ∂u ∂x ) + ∂ ∂x β(x)θ = f(x, t), (1) k(x) ∂θ ∂t + β(x) ∂u ∂x∂t − ∂ ∂x ( κ (x) ∂θ ∂x ) = φ(x, t), (2) where ρε is the mass density of string, α is a stiffness coefficient, k is the specific heat, κ denotes the thermal conductivity coefficient, and β is a coupling coefficient; f and φ represent an external force and heat source, respectively. For a finite string, it is not restrictive to assume that the reference configuration is the interval (a, b) with a < 0 < b. We are interested in local perturbations of the mass density, ρε(x) = { p(x) if x ∈ (a,−ε) ∪ (ε, b) ε−mq ( x ε ) if x ∈ (−ε, ε) , represented by two parameters ε → 0 and m ∈ R. The mass density functions p : [a, b] → R and q : [−1, 1] → R are bounded and strictly positive. We assume that p is continuous on [a, 0) and (0, b], while q is continuous on [−1, 1]. We also suppose that all other parameters are strictly positive in Ω̄ and smooth enough, namely α, β,κ ∈ C(a, b), and k ∈ C(a, b). In this note, we analyse the asymptotic behaviour as ε → 0 of the eigenvalues λε and eigenvectors (uε, θε) of the eigenvalue problem associated with (1) and (2) on Ωε := (a,−ε) ∪ (−ε, ε) ∪ (ε, b), − (α(x)uε) ′ + (β(x)θε) ′ = −λερε(x)uε, (3) − (κ (x)θ′ ε) ′ − λεβ(x)uε = λεk(x)θε, (4) complemented with the Dirichlet boundary conditions at the outer ends uε(a) = uε(b) = 0 and θε(a) = θε(b) = 0, (5)