  The mixing of active neutrinos with their sterile counterparts with keV mass
is known to have a potentially major impact on the energy loss from the
supernova core. By relying on a set of three static hydrodynamical backgrounds
mimicking the early accretion phase and the Kelvin-Helmoltz cooling phase of a
supernova, we develop the first self-consistent, radial- and time-dependent
treatment of sterile and tau neutrinos mixing in the dense stellar core. We
follow the flavor evolution by including ordinary matter effects, collisional
production of sterile neutrinos, as well as reconversions of sterile states
into active ones. The dynamical feedback of the sterile neutrino production on
the matter background leads to the development of a tau lepton asymmetry that
grows in time until it reaches a value larger than 0.15. Our results hint
towards significant implications for the supernova physics, and call for a
self-consistent modeling of the sterile neutrino transport in the supernova
core to constrain the mixing parameters of sterile neutrinos.
