One option is to have another abstract subclass, to use as the superclass for all implementations which do want to use the default implementation.
Personally I usually leave non-final methods abstract in an abstract class (or just use interfaces instead) but it definitely depends on the situation. If you have an interface with many methods, and you want to be able to just opt in to some of them, for example, then an abstract class which implements the interface in a no-op way for every method is fine.
You need to evaluate each case on its merits, basically.