  The observational data on ultrahigh energy cosmic rays (UHECR), in particular
their mass composition, show strong indications for extremely hard spectra of
individual mass groups of CR nuclei at Earth. In this work, we show that such
hard spectra can be the result of the finite life-time of UHECR sources, if a
few individual sources dominate the UHECR flux at the highest energies. In this
case, time delays induced by deflections in the turbulent extragalactic
magnetic field as well as from the diffusive or advective escape from the
source environment can suppress low-energy CRs, leading to a steepening of the
observed spectrum. Considering radio galaxies as the main source of UHECRs, we
discuss the necessary conditions that few individual sources dominate over the
total contribution from the bulk of sources that have been active in the past.
We provide two proof-of-principle scenarios showing that for a turbulent
extragalactic magnetic field with a strength $B$ and a coherence length $l_{\rm
coh}$, the life-time of a source at a distance $d_{\rm src}$ should satisfy
${t_{\rm act} \sim \left( B/1\,\text{nG} \right)^2\,\left( d_{\rm
src}/10\,\text{Mpc} \right)^2\,\left( l_{\rm coh}/1\,\text{Mpc}
\right)\,\text{Myr}}$ to obtain the necessary hardening of the CR spectrum at
Earth.
