  For the first time ever, we derive equations governing the time-evolution of
fully relativistic slim accretion disks in the Kerr metric, and numerically
construct their detailed non-stationary models. We discuss applications of
these general results to a possible limit-cycle behavior of thermally unstable
disks. Our equations and numerical method are applicable in a wide class of
possible viscosity prescriptions, but in this paper we use a diffusive form of
the "standard alpha prescription" that assumes the viscous torque is
proportional to the total pressure. In this particular case, we find that the
parameters which dominate the limit-cycle properties are the mass-supply rate
and the value of the alpha-viscosity parameter. Although the duration of the
cycle (or the outburst) does not exhibit any clear dependence on the black hole
spin, the maximal outburst luminosity (in the Eddington units) is positively
correlated with the spin value. We suggest a simple method for a rough estimate
of the black hole spin based on the maximal luminosity and the ratio of
outburst to cycle durations. We also discuss a temperature-luminosity relation
for the Kerr black hole accretion discs limit-cycle. Based on these results we
discuss the limit-cycle behavior observed in microquasar GRS 1915+105. We also
extend this study to several non-standard viscosity prescriptions, including a
"delayed heating" prescription recently stimulated by the recent MHD
simulations of accretion disks.
