General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L1(0,∞) relaxation function g namely g′(t)≤−ξ(t)H(g(t)), where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=sp 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.