I adopted my stepson on the grounds of abandonment. The biological mother had stopped contacting my stepson, and she moved without notifying us or giving us a fowarding address. We had to file for the adoption through a public notice.
In our case there was also no child support order, my husband never asked for any, but that only helped our case in the end because there was no contribution on her part at all to my stepson's life.
Stepparent Adoption Blog