Optimal Stabilization for Nonlinear Abstract Time-Delayed Viscoelastic Equations With Infinite Memory

This paper is concerned with a class of abstract viscoelastic equations in which the delay element appears in the nonlinear internal damping and infinite memory is considered. We study the stability result of energy for a wide range of kernel functions and establish the decay estimates of the energy solution. The energy decay is more general and explicit under appropriate conditions for both delayed and nondelayed frictional dampings by introducing an appropriate Lyapunov function and some arguments of convex functions. Finally, a semilinear wave system as an application and some examples of energy decay rates are presented.