  Ultra light axion fields, motivated by the string theory, form a large
condensate (axion cloud) around rotating black holes through superradiant
instability. Several effects due to the axion cloud, such as the spin-down of
black holes and the emission of monochromatic gravitational waves, open a new
window to search for axions by astrophysical observations. When the axion
self-interaction is considered, the evolution of cloud is altered
significantly, and an explosive phenomenon called bosenova can happen. Thus, it
is necessary to understand the precise evolution of self-interacting clouds for
the detection of axions by astrophysical observations. In this paper, we
propose a new method to track the whole process of the growth of
self-interacting axion clouds employing the adiabatic approximation. We
emphasize that our method relies neither on the non-relativistic approximation
nor on perturbative treatment of the self-interaction, which is often used in
literature. Our main finding is that the evolution of cloud in the strongly
self-interacting regime depends on the strength of the gravitational coupling
between the axion and the black hole. For a large coupling, the cloud evolves
into a quasi-stationary state where the superradiant energy gain is balanced
with the energy dissipation to infinity by the self-interaction. On the other
hand, when one decreases the size of coupling, clouds become unstable at some
energy, which would be interpreted as the onset of bosenova.
