Exponential Decay for a Wave Equation With a Discrete Time Delay

We present a novel approach for treating delay terms broadly, with a specific focus on wave equations. This method allows us to establish exponential energy decay in the presence of a time retardation and the concurrence of a strong damping. Notably, it reveals new types of interactions in which the time delay plays a central role. We show that for small delays, the energy decays exponentially–even when the damping is relatively “weak.” This work introduces a distinct method that adds to our prior contributions as well as the existing treatments of delayed wave equations in the literature.