My question involves name change laws in the State of: Ohio/Tennessee

My son is 2 1/2 and was born in Ohio. His father and I were never married and he was not placed on the BC when he was born since he disappeared when I was six months pregnant until my son was almost 1. I had paternity established and I put his father on his BC in Ohio. I then relocated to Tennessee two years ago. His father has married and moved to Tennessee and we are currently fighting for custody. He has visitation and pays support. My son has my last name. Can he change my son's name to his without my consent? Can I prevent him from changing my son's name?