  The survival probabilities of a ``prisoner'' diffusing in an expanding cage
and a ``daredevil'' diffusing at the edge of a receding cliff are investigated.
When the diffuser reaches the boundary, he dies. For ``marginal'' boundary
motion, i.e., the cage length grows as $\sqrt{At}$ or the cliff location
recedes as $x_0(t)=-\sqrt{At}$ and the daredevil diffuses within the domain
$x>x_0$, the survival probability of the diffuser exhibits non-universal
power-law behavior, $S(t)\sim t^{-\b}$, which depends on the relative rates of
boundary and diffuser motion. Heuristic approaches are applied for the cases of
``slow'' and ``fast'' boundary motion which yield approximate expressions for
$\b$. An asymptotically exact analysis of these two problems is also performed
and the approximate expressions for $\b$ coincide with the exact results for
nearly entire range of possible boundary motions.
