Going into a PhD expecting to take longer
My advice, for the US, is to share your medical condition as narrowly as possible. Most US universities will have an office for student support in such things and they may need to know so that they can provide some services that will help you. But I, personally, probably wouldn't share it even with an advisor unless and until it proves to be an actual issue. I don't know whether this is a purely US view, but there are rules about discrimination for medical conditions here.
I also wouldn't try to work out in advance some &quot;part time&quot; degree. Just take as long as you need to take (caveat below).
I'm guessing a bit here, but also suggest that you are better off in pure math than applied math here, since the former seldom has deadlines to worry about. Research takes as long as it takes. And you are probably better off in a subfield that you already know pretty well to help you get through any coursework and comprehensive exams.
Don't worry about or suggest &quot;partial stipends&quot;. Most doctoral students in the US serve as TAs, working with students in elementary classes, grading or running study sessions. Stipends without duties, as such, are pretty rare, though you seem to be writing about a pretty high level place where they may be a bit less rare. But they are likely funded by grants, even then.  If you can put in the time for that, then it doesn't need to end since you are providing a service that the university vitally needs.
Caveat: Some universities have a quite firm maximum number of years of study. Make sure you know about that before you begin, but also that you know what accommodation can be made in your case. And many students take seven years to complete for other reasons. You don't have to hit the median to be successful.
Again, the student support office can be your friend.