  The hypothesis of the decay of neutrino mass eigenstates leads to a
substantial modification of the appearance and disappearance probabilities of
flavor eigenstates. We investigate the impact on the standard oscillation
scenario caused by the decay of the heaviest mass eigenstate $\nu_3$ (with a
mass $m_3$ and a mean life $\tau_3$) to a sterile state in DUNE. We find that
the lower bound of $5.1 \times 10^{-11}~s/eV$ at 90\% CL on the decay parameter
$\tau_3/m_3$ can be set if the Neutral Current data are included in the
analysis, thus providing the best long-baseline expected limit so far. We also
show that the $\nu_\tau$ appearance channel would give only a negligible
contribution to the decay parameter constraints. Our numerical results are
corroborated by analytical formulae for the appearance and disappearance
probabilities in vacuum (which is a useful approximation for the study of the
invisible decay model) that we have developed up to the second order in the
solar mass splitting and to all orders in the decay factor $t/\tau_3$.
