This is a very legitimate question, but asked in a wrong way. The real question has nothing to do with DX versus FX. It's what will be the impact of mirrorless camera technology on DX? To be more specific, it's how large a chunk of the DX market will be grabbed by mirrorless cameras with electronic viewfinders (EVF) and continuous focus (CF)?
Basically, dSLRs with optical viewfinders and phase detection focusing don't integrate well with video, while mirrorless cameras do. While the technology is just getting going, over the next year, there should be at least three companies with mirrorless cameras that flow seamlessly between stills and video, and also work fast enough to be considered competent. Panasonic's GH2 is the first and Sony seems headed that way with its not-yet-announced NEX-7. Nikon will also likely make the cut with its new EVIL, assuming it is geared towards the high end of the market.
Initially, these three will have higher end cameras with kits starting above $1K, but the technology will then drop in price rapidly, so that you'll have competition at every market segment. Will DX die? Certainly not initially. But if the mirrorless camera manufacturers can match DX still image quality and provide a similar ergonomic experience to optical viewfinders with EVF, then they will likely take a huge chunk out of DX sales.
Note, incidentally, that you can certainly expect both Nikon and Canon to hedge their bets with both formats.