Second Order Linear Evolution Equations with General Dissipation

The contraction semigroup S(t) = eᵗᴬ generated by the abstract linear dissipative evolution equation u¨+Au+f(A)u˙=0is analyzed, where A is a strictly positive selfadjoint operator and f is an arbitrary nonnegative continuous function on the spectrum of A. A full description of the spectrum of the infinitesimal generator A of S(t) is provided. Necessary and sufficient conditions for the stability, the semiuniform 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 also discussed.