When Did The Shakespeare Family Name Die?

Here’s a question that occurred to me the other day, and I don’t know the answer.  Shakespeare had a son, Hamnet, who died as a child.  He also had two daughters, Susana and Judith, who got married and had families of their own.  Interestingly, Judith named one of her children “Shakespeare” as a first name(*), which leads to my question… What happened to the Shakespeare family name?  How long did it last after William, since he had no heir to pass it down? A little bit of searching turns up Shakespeare’s Family Tree which shows that although Shakespeare did have several brothers (Gilbert, Richard and Edmond), it appears that none of them married.  His sister Joan married, but again, same problem, what happens to the Shakespeare family name?  Was William it?    (*) I know at least one couple who had a similar issue – she was an only child, and while she did want to take her husband’s name, she wanted to do something to recognize her father’s family name.  So she posed the idea of having her current last name be the baby’s first, or at least middle, name.  They didn’t go through with it.  Judith’s Shakespeare, by the way, was her first born, who apparently only lived one year.  Sad, but common in the day.  Don’t worry, Judith.  We remember your dad’s name.