In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log　terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce two partial generalizations of the result by Birkar--Cascini--Hacon--McKernan. We also discuss some applications of the generalizations. This is a joint work with Zheng-Yu Hu.