Well-posedness and Stability of Two Classes of Plate Equations with Memory and Strong Time-dependent Delay

Abstract

Two classes of plate equations with past history and strong time-dependent delay in the internal feedback are considered. Our results contain the global well-posedness and exponential stability of the two systems. We prove the global well-posedness of a system with rotational inertia without any restrictions on $\mu_1$, $\mu_2$, and the system without rotational inertia under the assumption $|\mu_2| \leq \mu_1$. For the system with rotational inertia, we establish exponential stability to the plate equation with the memory term only to control the delay term if the amplitude of the time delay term is small, and the stability result also holds for the plate equation with strong anti-damping. For the system without rotational inertia, we obtain the exponential stability under the assumption $|\mu_2| \lt \sqrt{1-d} \mu_1$.