In this talk new enclosures for the spectra of operators associated with second order Cauchy problems are presented for non-selfadjoint damping. Our new results yield much better bounds than the numerical range of these non-selfadjoint operators for both uniformly accretive and sectorial damping.
(joint work with B. Jacob, Carsten Trunk and H. Vogt)

General theory of operator semigroups provides abstract results which can be used to obtain optimal rates of decay or convergence in many evolution equations or dynamical systems. I will describe the abstract results, and indicate how they are obtained and how they can be applied in examples.