A fake-up is when a couple says, often in order to increase the dramatic effect of an argument, that they are going to break up, when in fact neither of them believes that they will. It can also be when a frequently "on-off" couple swears they're broken up for good this time, and then gets back together.
Person A: Did you hear about Ken and Barbie's fight?
Person B: Yeah, I heard she threatened to break up with him, but he knew she wouldn't actually go through with it.
Person A: Yeah, she's always trying to fake-up with that boy...