  The double-degenerate model, involving the merger of double carbon-oxygen
white dwarfs (CO WDs), is one of the two classic models for the progenitors of
type Ia supernovae (SNe Ia). Previous studies suggested that off-centre carbon
burning would occur if the mass-accretion rate (Macc) is relatively high during
the merging process, leading to the formation of oxygen-neon (ONe) cores that
may collapse into neutron stars. However, the off-centre carbon burning is
still incompletely understood, especially when the inwardly propagating burning
wave reaches the centre. In this paper, we aim to investigate the propagating
characteristics of burning waves and the subsequently evolutionary outcomes of
these CO cores. We simulated the long-term evolution of CO WDs that accrete
CO-rich material by employing the stellar evolution code MESA on the basis of
the thick-disc assumption. We found that the final outcomes of CO WDs strongly
depend on Macc (Msun/yr) based on the thick-disc assumption, which can be
divided into four regions: (1) explosive carbon ignition in the centre, then
SNe Ia (Macc < 2.45*10^-6); (2) OSi cores, then neutron stars (2.45*10^-6 <
Macc < 4.5*10^-6); (3) ONe cores, then e-capture SNe (4.5*10^-6 < Macc <
1.05*10^-5); (4) off-centre oxygen and neon ignition, then off-centre explosion
or Si-Fe cores (Macc > 1.05*10^-5). Our results indicate that the final fates
of double CO WD mergers are strongly dependent on the merging processes (e.g.
slow merger, fast merger, composite merger, violent merger, etc.).
