On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction

We investigate the stability of three thermoelastic beam systems with hyperbolic heat conduction. First, we study Bresse-Gurtin-Pipkin system, providing a necessary and sufficient condition for exponential optimal polynomial decay rate when is violated. Second, obtain analogous results Bresse-Maxwell-Cattaneo completing an analysis recently initiated in literature. Finally, consider Timoshenko-Gurtin-Pipkin system find known does not hold. As byproduct, fully recover characterization Timoshenko-Maxwell-Cattaneo system. The classical "equal wave speeds" conditions are also recovered through singular limit procedures. Our compatible some physical constraints on coefficients as positivity Poisson's ratio material. faces several challenges connected thermal damping, whose resolution rests developed mathematical tools such quantitative Riemann-Lebesgue lemmas.