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 σ-evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.