  Galaxy clusters are promising targets for indirect dark matter searches.
Gamma-ray signatures from the decay or annihilation of dark matter particles
inside these clusters could be observable with the Fermi Large Area Telescope
(LAT). Based on three years of Fermi LAT gamma-ray data, we analyze the flux
coming from eight nearby clusters individually as well as in a combined
likelihood analysis. Concentrating mostly on signals from dark matter decay, we
take into account uncertainties of the cluster masses as determined by X-ray
observations and model the cluster emission as extended sources. Searching for
different hadronic and leptonic decay and annihilation spectra, we do not find
significant emission from any of the considered clusters and present limits on
the dark matter lifetime and annihilation cross-section. We compare our
lifetime limits derived from cluster observations with the limits that can be
obtained from the extragalactic gamma-ray background (EGBG), and find that in
case of hadronic decay the cluster limits become competitive at dark matter
masses below a few hundred GeV. In case of leptonic decay, however, galaxy
cluster limits are stronger than the limits from the EGBG over the full
considered mass range. Finally, we show that in presence of dark matter
substructures down to 10^-6 solar masses the limits on the dark matter
annihilation cross-section could improve by a factor of a few hundred, possibly
going down to the thermal cross-section of 3 10^-26 cm^3 s^-1 for dark matter
masses < 150 GeV and annihilation into $b\bar{b}$. As a direct application of
our results, we derive limits on the lifetime of gravitino dark matter in
scenarios with R-parity violation. Implications of these limits for the
possible observation of long-lived superparticles at the LHC are discussed.
