Stability of a Degenerate Thermoelastic Equation

This work is dedicated to the study of a linear model arising in a thermoelastic rod of homogeneous material. The system results 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 endpoint 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 use a frequency domain approach based on the well-known result of Prüss in order to prove, using some multiplier techniques, that the energy of classical solutions decays uniformly as time goes to infinity.