The only way DX will go away is because Nikon chose to go FX only. Or that they can't source the DX sensors.
Nikon's push towards FX is because there is more profit in FX cameras, and they see it as a sector for growth in a contracting camera market. And they probably believe that any up-graders from DX, or another brand, will start buying new lenses for their new format, so their is more money to be had from the new FX users which there may not be from new DX users, or at least not to the same extent.
They run the risk of alienating their DX users who don't want to go the FX route, but want options for lenses optimised for their system, and/or an upgrade path to the top of a DX range with 'Pro' features. But the vast majority of Nikon users are DX users of their D3*** and D5*** range who may be happy with their camera and the 1-3 lenses.
And we are getting to a point where a lot of people, especially in the mid to lower range, have a good enough system for whatever they want to do, and so a slow down in growth is inevitable. For most people, they have enough pixels, so selling cameras by the number of pixels is becoming less important. Canon seem to have decided to have their APS-C cameras stay in 18Mp, and sell the cameras on other features. The 7DII could be the start of a change to that, time will tell, but it is not the huge jumps from 18 > 20Mp. Not like Nikon who went from 12 > 16 > 24Mp increases over the few years.
Obviously DX is only the Nikon name for their version of the APS-C size of sensor, and they are not the only company using that size of sensor. Everyone who sells DSLRs have a crop sensor format, and is normally where most, or all, of their money comes from for their DSLRs. Whilst many CSC cameras started with the M43 sensor, there are some using the APS-C size sensors, which will extend the the life of the format probably indefinitely.
For me, the DX format is the perfect sensor size, and long may it stick around.