  In this work we study the effects of interactions and disorder on $1D$ DIII
topological superconductors and the Majorana Kramers doublets (MKDs). In
contract to the case without the time-reversal symmetry, the Umklapp
interaction plays important roles in this system. The underlying phases due to
the Umklapp interaction and disorder are found by using a perturbative
renormaliztion analysis based on the Abelian Bosonization. Importantly, the
stable topological regime can be found within a rather wide parameter space.
Furthermore, the degeneracy splitting of the MKDs is shown to be still
exponentially dependent on the length of the wire in the presence of both the
Umklapp interaction and disorder, when the Luttinger parameter
$K_0>\sqrt{2}/2$. The differences caused by the Umklapp interaction are
highlighted in contrast to the time-reversal breaking cases.
