We could use a little history to help with the diagnosis.
About how long have you lived there and has this area always been different as we see it now.
Tug on some of the grass to see if it pulls easy. If it does grubs have cut the roots.
Soil - all soil doesn't retain moisture the same. To your knowledge was this area ever filled in with soil from another area?
Great question. I moved here about three years ago. Since the seller left the house on market for about a year the entire front lawn is pretty much either brown or covered with weed. I applied weed killer in spring and planted seeds in the fall. The next year condition has improved but the left part is still problematic. I remember at one point the mowing guy came then left and told me there was no point to mow a desert. In Fall I topdressed and overseeded the entire front yard (took me two days to moved about 3 cubic yards of soil with a shovel, lol!), but hard work paid off, a month later the entire yard turned green! I pretty much had the best looking lawn in the neighborhood in October. Then came last winter, the lawn was covered with snow for about 4 months. When the snow finally melted away I discovered that much of the grass on the left side didn't grow back, I am not sure if the cold has killed it or not. That's what I know about this area from the very beginning.
This fall I am going to focus on just the left part. I am pretty sure I will get good results going into the winter but I am afraid the same thing will happen next spring.
How deep should I topdress? half an inch?