New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function k_i, namely, ki′(t)≤−ξi(t)Ψi(ki(t)),i=1,2. We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when k_i(s) = s^p and p covers the full admissible range [1 , 2).