Well-posedness and Long-Time Behavior for a Degenerate Thermoelastic Euler–Bernoulli Beam Model

In this paper, we study the well-posedness and the long-time behavior of a degenerate thermoelastic Euler–Bernoulli beam system. The model consists of a fourth-order hyperbolic equation for the vertical deflection of the beam coupled with a second-order parabolic equation for its temperature evolution, both characterized by a space-dependent degenerate coefficient affecting the flexural rigidity and thermal diffusion. We first establish the well-posedness of the system in a suitable weighted energy Hilbert space via the Lumer–Phillips theorem. We then introduce an appropriate Lyapunov functional and make use of a special Hardy–Poincaré inequality to prove the uniform exponential stability of the system under a certain condition on the degeneracy.