A general decay result for a viscoelastic equation in the presence of past and finite history memories

In this work, we are concerned with the following equation utt−Δu+∫0tg1(t−s)div(a1(x)∇u(s))ds+∫0+∞g2(s)div(a2(x)∇u(t−s))ds=0 in a bounded domain ΩΩ. Under suitable conditions on a1a1 and a2a2 and for a wide class of relaxation functions g1g1 and g2g2, we establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases.