I'm in the following situation: currently I'm finishing my Master's in (mathematical) Logic in Amsterdam. I'm interested in doing a PhD (preferably in the US), either in something more applied or in continuation of my current research (set theory/proof theory). I've understood this logic department has excellent placement for PhDs (many go on to places like CMU, Oxbridge, Berkeley, Stanford) and the study program is considered fairly intensive. However, I have significant chronic health issues preventing me to study much beyond 70% of what is considered ordinary full time studies here, without it affecting my mental health (although I should add that in this program around 25-40% of the people don't fully reach the 100% rate). The illness does not affect my ability to do mathematics, just how much I can do effectively of it in a day. Therefore, in all likelihood I would need more time, or at least the possibility of more time, to finish a PhD successfully. Say the average time of some PhD program X to finish is 5 years, I might need 6.5-7 years. I've read many questions on this site related to my situation, but I get very mixed signals in the answers.
Given the situation is it even reasonable to attempt a PhD?
Suppose I were to apply for a PhD in (say) the US, and I am honest and upfront about the situation to my potential supervisor, would such extensions be possible? Or would it just mean an instant rejection?
If I'm open to pursue more applied avenues (logic in computer science perhaps) would this change the answer to the above two questions? The reason I ask is that I've gotten the strong impression that such programs are in general somewhat less intensive than those in pure mathematics. But this can also just be a symptom of superiority feeling that floats around in many pure math departments ;)