There'll be some level of support for a few years at least. Most of the major developers will swap over within a year of the launch (of whichever Sony/Microsoft console comes first) though, so within a short time there won't be a huge amount of big-name games appearing and a lot of what comes out will be things like annual sports games and lower-budget releases.
Almost every console manufacturer has supported their previous console after releasing a new one, for at least a year or so. The only company I can think of that abruptly ceased all support for their console after releasing the new one was Sega. When they released the Dreamcast they stopped supporting the Saturn right away and it pissed off a lot of third party developers, especially EA who refused to develop for the Dreamcast.