  This paper considers the neutron transport equation in bounded domain with a
combination of the diffusive boundary condition and the in-flow boundary
condition. We firstly study the existence of solution in any fixed time by
$L^2-L^{\infty}$ method, which was established to study Boltzmann equation in
\cite{[Guo2]}. Based on the uniform estimates of the solution, we also consider
the BV-regularity of the solution in non-convex domain. A cut-off function,
which aims to exclude all the characteristics emanating from the grazing set
$\mathfrak{S}_B$, has been constructed precisely.
