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There is really no way to eliminate him from their lives.
The Court is not at all likely going to sever all access rights. It is virtually never done unless there is a situation of danger so at this point in time it is not really possible. You could ask for an order for sole custody and no access and if he doesn't contest it then the court would grant the order but otherwise it is just not likely at this time. As time goes on though and years pass and he's not seen the children and then wants to, then you may be able to stop him.
If you are given an order for sole custody and no access you can change the children's last names. If you don't get that order or the father's permission then you need a court order allowing you to do so.
If you can show that he is avoiding service then your lawyer can get an order for substitute service so that you will be able to serve in in an alternative way such as to his last known address, to his family, by advertising etc.
So you can have the process server swear an affidavit showing that it was not possible to serve him and that will work. There is no time limit or number of times for some cut-off. It really comes down to showing the court you did what you could and he is avoiding service.
You are very welcome.
I am sorry you are in this terrible situation. I hope it works out quickly and smoothly.
I am saddened to hear this. I know you are right and your poor little boys will hope for a long time. I understand completely, believe me.
Have a good night.