  Within a concept of the fractional diffusion equation and subordination, the
paper examines the influence of a competition between long waiting times and
long jumps on the escape from the potential well. Applying analytical arguments
and numerical methods, we demonstrate that the presence of long waiting times
distributed according to a power-law distribution with a diverging mean leads
to very general asymptotic properties of the survival probability. The observed
survival probability asymptotically decays like a power-law whose form is not
affected by the value of the exponent characterizing the power-law jump length
distribution. It is demonstrated that this behavior is typical of and generic
for systems exhibiting long waiting times. We also show that the survival
probability has a universal character not only asymptotically but also at small
times. Finally, it is indicated which properties of the first passage time
density are sensitive to the exact value of the exponent characterizing the
jump length distribution.
