Why are my cabinets peeling and how do I fix it?
I'd recommend cleaning it up a bit, making sure it's totally dry and remove all the flakes of particle board that come off as you just rub it. Then treat it with some rotten wood stabilizer and sand the surface once dry. Clean the veneer really well, if you can without damaging it, lightly sand it.

Then apply contact cement to both surfaces and let dry.

Once dry, work from the point where the veneer began to peel towards the end that's totally detached. A j-roller won't be able to get into the tight curves, but perhaps you can use one around the outside edge. Just work it inward, applying lots of force as you press the two together. Go a little bit at a time so that you don't cause wrinkles. Use lots of force as you press down!

This should keep it together quite well.

The thing is, it'll happen to other cabinets and you'll waste a lot of time fixing this problem. Alternatively, you can replace them with solid wood doors (which are expensive) or even replace them with other crappy doors for relatively cheap that will last 5-10 years max. Ikea sells their doors separately, see here: http://www.ikea.com/us/en/catalog/categories/departments/kitchen/11479/