They'll stop, or at least slow down when the middle eastern governments begin showing zero tolerance for terrorist organizations within their countries. If we make that an imperative for the survival of said governments they'll have little choice. We can do this only with determined resolve to go the distance. Anything less will lead only to increased aggression.
I'm confident that Bush knew full well he was starting something that he couldn't finish, at least in one term. By "starting something" I don't mean he started this war. This war was begun long ago but escalated on 9-11-2001. What Bush started was a response to the Islamic fundamentalist aggression. A response that can only be successful when fully completed.
As divided as we Americans seem to be over so many things, I have strong doubts that we'll have the resolve to win this "war against terrorism". However, that we have a leader that is willing to risk very possible failure to do what must be done gives me hope.