  Based on the R-parity violation option of the minimal supersymmetric Standard
Model, we examine the scenario where the massive gravitino, relic from the hot
big-bang, is the lightest supersymmetric particle and can decay through one or
several of the trilinear R-parity violating interactions. We calculate the
rates of the gravitino decay via the various three-body decay channels with
final states involving three quarks and/or leptons. By taking into account the
present constraints on the trilinear R-parity violating coupling constants and
assuming the gravitino and scalar superpartner masses do not exceed $\sim 80
TeV$, it turns out that the gravitinos could easily have decayed before the
present epoch but not earlier than the big-bang nucleosynthesis one. Therefore,
the considered scenario would upset the standard big-bang nucleosynthesis and
we conclude that it does not seem to constitute a natural solution for the
cosmological gravitino problem.
