General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L-1 (0, infinity) relaxation function g namely g'(t) <=-xi(t)H(g(t)), where H is an increasing and convex function near the origin and xi 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) = 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. (c) 2017 Elsevier Inc. All rights reserved.