  In many random search processes of interest in chemistry, biology or during
rescue operations, an entity must find a specific target site before the latter
becomes inactive, no longer available for reaction or lost. We present exact
results on a minimal model system, a one-dimensional searcher performing a
discrete time random walk or L\'evy flight. In contrast with the case of a
permanent target, the capture probability and the conditional mean first
passage time can be optimized. The optimal L\'evy index takes a non-trivial
value, even in the long lifetime limit, and exhibits an abrupt transition as
the initial distance to the target is varied. Depending on the target lifetime,
this transition is discontinuous or continuous, separated by a non-conventional
tricritical point. These results pave the way to the optimization of search
processes under time constraints.
