On the asymptotic behavior of the energy for evolution models with oscillating time-dependent damping

In the present paper, we study the influence of oscillations of the time-dependent damping term b(t)u(t) on the asymptotic behavior of the energy for solutions to the Cauchy problem for a s-evolution equation {u(tt) + (- Delta)(sigma) u + b(t)u(t) = 0, (t, x) is an element of [0,for all) x R-n, u(0, x) = u(0)(x), ut (0, x) = u(1)(x), x is an element of R-n, where sigma > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b(t) = beta(1 + t)(-1), with beta > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of beta.