Optimal decay rates for the viscoelastic wave equation

In this paper, we consider a viscoelastic equation with minimal conditions on the L1(0,) relaxation function g, namely, g(t)-(t)H(g(t)), where H is an increasing and convex function near the origin and is a nonincreasing function. With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=s(p) and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.