When is it child abandonment?

My fiance' has raised my son his whole life. His biological father left when he was a few mos old and my fiance has been there since he was 4 mos (he is now 4yrs). My fiance wants to adopt him and give him our last name but I don't know how to find my sons bio father. If I did he would never agree to it anyways. Can I still do it without consent due to abandonment?