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

Abstract

In the present paper, we study the influence of oscillations of the time-dependent damping term $\mathit{b}\mathrm{(}\mathit{t}\mathrm{)}{\mathit{u}}_{\mathit{t}}$ on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation $$\begin{array}{c}\left\{\begin{array}{c}{\mathit{u}}_{\mathit{t}\mathit{t}}+{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{\sigma }}\mathit{u}+\mathit{b}\mathrm{(}\mathit{t}\mathrm{)}{\mathit{u}}_{\mathit{t}}=0\mathrm{,}\mathrm{(}\mathit{t}\mathrm{,}\mathit{x}\mathrm{)}\in [0\mathrm{,}\infty \mathrm{)}\times {\mathbb{R}}^{\mathit{n}}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}0\mathrm{,}\mathit{x}\mathrm{)}={\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}{\mathit{u}}_{\mathit{t}}\mathrm{(}0\mathrm{,}\mathit{x}\mathrm{)}={\mathit{u}}_1\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{x}\in {\mathbb{R}}^{\mathit{n}}\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{\sigma }\mathrm{>}0$ and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case $\mathit{b}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\beta }{\mathrm{(}1+\mathit{t}\mathrm{)}}^{-1}$ , with $\mathit{\beta }\mathrm{>}0$ , and we discuss the influence of oscillations of b on the energy estimates according to the size of β.