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 $$\ddot{u}+Au+f(A)\dot{u}=0$$ is analyzed, where $A$ is a strictly positive selfadjoint operator and $f$ is an arbitrary nonnegative continuous function on the spectrum of $A$. A 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.