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The contraction semigroup $S(t)={\rm e}^{t\mathbb{A}}$ generated by theabstract linear dissipative evolution equation $$ \ddot u + A u + f(A) \dot u=0$$ is analyzed, where $A$ is a strictly positive selfadjoint operator and $f$is an arbitrary nonnegative continuous function on the spectrum of $A$. A fulldescription of the spectrum of the infinitesimal generator $\mathbb{A}$ of$S(t)$ is provided. Necessary and sufficient conditions for the stability, thesemiuniform stability and the exponential stability of the semigroup are found,depending on the behavior of $f$ and the spectral properties of its zero-set.Applications to wave, beam and plate equations with fractional damping are alsodiscussed.

Second order linear evolution equations with general dissipation