  In this work we study the asymptotic behavior of solutions for a general
linear second-order evolution differential equation in time with fractional
Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with
less demand on the initial data when compared to previous results in the
literature. In certain cases, we observe that the dissipative structure of the
equation is of regularity-loss type. Due to that special structure, to get
decay estimates in high frequency region in the Fourier space it is necessary
to impose additional regularity on the initial data to obtain the same decay
estimates as in low frequency region. The results obtained in this work can be
applied to several initial value problems associated to second-order equations,
as for example, wave equation, plate equation, IBq, among others.
