Second Order Linear Evolution Equations with General Dissipation

Abstract

The contraction semigroup $$S(t)=\mathrm{e}^{t\mathbb {A}}$$ generated by the abstract linear dissipative evolution equation $$\begin{aligned} \ddot{u} + A u + f(A) \dot{u}=0 \end{aligned}$$ is 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 $$\mathbb {A}$$ 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.