Basement ceiling

I'm finishing my 50 year old basement and need advice regarding the 6'x4' ceiling area (which is actually a cement slab from the front porch). It drops down 2" lower than the rest of the ceiling. The rest of the basement only has 6'9" clearance so lowering the main ceiling to match this area is out of the question.

Would you seal this or just install a thermal barrier? And what kind of barrier? Any suggestions on how to attach the barrier and the drywall? I'm trying to avoid drilling into the cement as much as possible.

I hope this is now concise enough to meet forum rules.

Thanks in advance.