  The decay of neutrino mass eigenstates leads to a change of the conversion
and survival probability of neutrino flavor eigenstates. Remarkably, we find
that the neutrino decay provides an enhancement of the expected tau appearance
signal with respect to the standard oscillation scenario for the long-baseline
OPERA experiment. The increase of the $\nu_\mu \rightarrow \nu_\tau$ conversion
probability by the decay of one of the mass eigenstates is due to a reduction
of the "destructive interference" among the different massive neutrino
components. Motivated by the recently released results of the OPERA
Collaboration showing a number of observed $\nu_\tau$ events larger than
expected, we perform a statistical analysis including the invisible decay
hypothesis. We obtain a very mild preference for invisible decays, with a best
fit value $\tau_3/m_3 \simeq 2.6\times 10^{-13} $ s/eV, and a constraint at the
90$\%$ confidence level for the neutrino decay lifetime to be $\tau_3/m_3
\gtrsim 1.3\times 10^{-13}$ s/eV.
