  The carbon-oxygen white dwarf (CO WD) + He star channel is one of the
promising ways for producing type Ia supernovae (SNe Ia) with short delay
times. Recent studies found that carbon under the He-shell can be ignited if
the mass-accretion rate of CO WD is higher than a critical rate (about 2 *
10^-6 Msun/yr), triggering an inwardly propagating carbon flame. Previous
studies usually supposed that the off-centre carbon flame would reach the
centre, resulting in the formation of an oxygen-neon (ONe) WD that will
collapse into a neutron star. However, the process of off-centre carbon burning
is not well studied. This may result in some uncertainties on the final fates
of CO WDs. By employing MESA, we simulated the long-term evolution of
off-centre carbon burning in He-accreting CO WDs. We found that the inwardly
propagating carbon flame transforms the CO WDs into OSi cores directly but not
ONe cores owing to the high temperature of the burning front. We suggest that
the final fates of the CO WDs may be OSi WDs under the conditions of off-centre
carbon burning, or explode as iron-core-collapse SNe if the mass-accretion
continues. We also found that the mass-fractions of silicon in the OSi cores
are sensitive to the mass-accretion rates.
