General stability of a triple layer beam with time-varying delay and weak internal damping

Analyzing a triple layer Rao-Nakra beam model which has time-dependent weight function and delay, weak internal damping, with thermal dissipation elements, is our focus. Heat conduction from Fourier's law was adopted to regulate the thermal dissipation. It is known that when the three equations of the Rao-Nakra beam is directly and globally damped, an exponential stability is achieved. Weak internal damping leads to prolonged resonance, extended vibrational amplitudes and periods, as well as delayed energy dissipation. In this work we will rigorously demonstrate that despite weak internal damping and delay term in the bottom layer's equation, we can obtain a general stability for the Rao-Nakra beam model. As a result, exponential and polynomial decay can be deduced as particular instances.