  We study the survival probability of moving relativistic unstable particles
with definite momentum $\vec{p} \neq 0$. The amplitude of the survival
probability of these particles is calculated using its integral representation.
We found decay curves of such particles for the quantum mechanical models
considered. These model studies show that late time deviations of the survival
probability of these particles from the exponential form of the decay law, that
is the transition times region between exponential and non-expo\-nen\-tial form
of the survival probability, should occur much earlier than it follows from the
classical standard approach resolving itself into replacing time $t$ by
$t/\gamma$ (where $\gamma$ is the relativistic Lorentz factor) in the formula
for the survival probability and that the survival probabilities should tend to
zero as $t\rightarrow \infty$ much slower than one would expect using classical
time dilation relation. Here we show also that for some physically admissible
models of unstable states the computed decay curves of the moving particles
have fluctuating form at relatively short times including times of order of the
lifetime.
