There are a couple of options here, depending on how in-depth you want to get.
Option 1: Name Change - your mother can file an application with the court to change your last name to be the same as your step-dad. The court will set a hearing and your mother must arrange to serve your biological father with notice of that hearing so that he can show up to object if he disagrees that your name should be changed. IF he shows up, the judge would make a decision about whether it is in your best interests to change your name (you would get to testify). If he does NOT show up, the judge is likely to grant the name change in his absence.
Option 2: Your mother can consider taking action to terminate your father's rights and allow your step-father to adopt you. This can happen the "easy" way (your father consents to the adoption) or the "hard" way (your mother files a separate action to terminate his rights on the grounds of abandonment, and either he no-shows, so it is granted, or he shows up and there is a trial on whether it should be granted).
Another option is for you to wait until you turn 18 and then you can file for your own name change (you do not need to contact your father if you are an adult), or you can even file to have your step-dad adopt you as an adult.
We offer a free consultation, so please feel free to have your mother contact us to schedule an appointment.
** This response is provided as general information only and does not constitute legal advice or otherwise establish an attorney client relationship. **Ask a similar question
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