Decay Rates of the Energy for an Abstract Viscoelastic Evolution Equation With Nonlinear Time-Dependent Damping

An abstract viscoelastic evolution equation with nonlinear time-dependent damping is considered in a Hilbert space setting. First, the well-posedness of solutions is proved by exploiting the semigroup theory. By establishing a weighted integral inequality, the decay of the energy is obtained under the condition that the memory kernel μ satisfies μ′(t)≤−ξ(t)μ(t) for t≥0, where ξ:[0,∞)→ℝ+ is a nonincreasing differentiable function. We derive results that are independent of ξ(t) and the coefficient γ(t) of the nonlinear time-dependent damping.