General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the $L^1(0,\infty )$ relaxation function g namely $g^{\prime }(t)\le ?\xi (t)H(g(t))$, 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 $H(s)=s^p$ 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.