Second sound thermoelastic stability of a string/beam structure

TL;DR

This paper investigates the stability of a coupled string/beam structure with thermoelastic components, demonstrating exponential decay when the string is thermoelastic and polynomial decay when only the beam is thermoelastic, using Cattaneo's law.

Contribution

It provides the first analysis of thermoelastic stability in a coupled string/beam system with Cattaneo's law, revealing different decay behaviors based on which component is thermoelastic.

Findings

Exponential energy decay when the string is thermoelastic.

Polynomial decay of order 1/t when only the beam is thermoelastic.

Decay rate can be at most polynomially stable of order 1/t^2.

Abstract

In this paper we study the one dimensional thermoelastic transmission problem in a special string/beam structure: the two components are coupled at an interface (identified to $0$). Either the string or the beam is supposed thermoelastic, the heat flux is given by the Cattaneo's law instead of the usual Fourier's law. We prove that the the energy decay of the whole system is exponential if the string is thermoelastic. When only the beam is thermoelastic, we prove that the energy of the coupling string/beam decays polynomially to zero as $\frac{1}{t}$ and the decay rate can be, at most, polynomially stable of order $\frac{1}{t^2}$.