Sure, competitors will start to gain more traction, but they're already in a lot of markets and competitors can't afford to have the same coverage as Uber. So while they'll lose a lot of people that are only local (see Austin for an example of this happening in a more complete way), I think they will retain pretty much all international/business travel type business, which is a pretty big moneymaker, just due to the convenience of "one app".
e.g. when I fly from London to the states, I take an Uber to the airport in London and take an Uber from the airport wherever I land (assuming they have Uber). One app, one linked credit card, etc.
I don't want a different on-demand car app for each city I happen to visit, which is where Uber will win out for the foreseeable future.