Stability of a degenerate thermoelastic equation

TL;DR

This paper investigates the stability of a thermoelastic rod model with degenerate heat conduction, analyzing well-posedness and energy decay using semigroup theory and frequency domain methods for weak and strong degeneracy cases.

Contribution

It provides a rigorous analysis of the well-posedness and uniform energy decay for degenerate thermoelastic equations with different degeneracy levels.

Findings

Energy of solutions decays uniformly over time

Established well-posedness for both weak and strong degeneracy models

Applied frequency domain techniques to prove stability

Abstract

This work is dedicated to the study of a linear model arising in thermoelastic rod of homogeneous material. The system is resulting from a coupling of a heat and a wave equation in the interval $(0,1)$ with Dirichlet boundary conditions at the outer endpoints where the parabolic component is degenerating at the end point $x=0$. Two models are considered the first is with weak degeneracy and the second is with strong degeneracy. We aim to study the well-posedness and asymptotic stability of both systems using techniques from the $C_{0}$-semigroup theory and a use a frequency domain approach based on the well-known result of Pr\"uss in order to prove using some multiplier techniques that the energy of classical solutions decays uniformly as time goes to infinity.