Tenant breakup

[Hilary Tetu]

 I have run into a difficult situation with my tenants in my first rental property. After running extensive screenings (background/credit checks, references from employers and landlords), the tenants have split up and are breaking their lease after a month and a half. They are paid up for rent for the month, the woman has moved out and the man and his children are still living there. He says that he can not afford to stay and will be out by the end of the month, but I am not convinced that he will actually move. I have expressed to him that he will be responsible for rent if he stays. My question is, do I advertise the house for rent starting next month? What if he isn't out by then? Is he then responsible for the entire month's rent or only half? Thanks for any insight.