Absence of exponential stability and polynomial stabilization in a class of beam models with tip rotary inertia

Abstract We investigate the impact of dissipative dynamic boundary conditions applied to one end of a beam and analyse their influence on the stability of the model within the Euler–Bernoulli framework. Our main finding is that hybrid dissipation does not alter the decay characteristics of the original model. We examine two scenarios: first, when hybrid dissipation is the sole dissipative mechanism; and second, when it complements other dissipative mechanisms. In the first case, we demonstrate that hybrid dissipation fails to induce exponential decay and instead produces a slow decay rate of $t^{-1/2}$ for large $t$. In the second case, when acting as a complementary mechanism, hybrid dissipation neither enhances nor diminishes the decay behavior of the original model.