I am in the process of filing for full custody of my son by publication. I want to know how long his father has to be gone in order to claim abandonment in Utah.

I don't think there is any such thing. If you are filing for full custody, and he's not around to contest it then you win by default. All you need to do is show a good faith effort to serve him with court papers. If he's around to contest it then he hasn't abandoned anyone.