  We consider the stationary metrics that have both the black hole and the
ergoregion. The class of such metric contains, in particular, the Kerr metric.
We study the Cauchy problem with highly oscillatory initial data supported in a
neighborhood inside the ergoregion with some initial energy $E_0$. We prove
that when the time variable $x_0$ increases this solution splits into two
parts: one with the negative energy $-E_1$ ending at the event horizon in a
finite time, and the second part, with the energy $E_2=E_0+E_1>E_0$, escaping,
under some conditions, to the infinity when $x_0\rightarrow +\infty$. Thus we
get the superradiance phenomenon. In the case of the Kerr metric the
superradiance phenomenon is "short-lived", since both the solutions with
positive and negative energies cross the outer event horizon in a finite time
(modulo $O(\frac{1}{k})$) where $k$ is a large parameter. We show that these
solutions end on the singularity ring in a finite time. We study also the case
of naked singularity.
