What can I do about peeling wok seasoning?
The most likely problem is that you let oil pool in the bottom of the wok during seasoning, so it didn't polymerize properly.  Personally, I would scrub that portion off, and reseason only the bottom, using this method:

Heat the wok to smoking hot.
Turn off the heat, and immediately polish with a lightly oiled cloth or paper towel.
Turn it upside down to cool (this prevents oil pooling at the bottom).

A second possibility is that you cooked something very wet, and maybe acidic as well, in the bottom of the wok while the seasoning was still fairly new.  Again, the answer is to scrub off the damaged portion and reseason it.
Some relevant links for general care of carbon steel woks:

Patina burning off
When to redo seasoning
Serious Eats wok seasoning

Overall, don't worry about it too much; I find that I mess up the seasoning in my wok (usually due to too much steaming) at least once a year and re-do it.  As long as you don't scratch up the metal (use a bamboo or plastic scrubby), you can rescue your wok from any kind of seasoning problem.