Last weekend the kids and I took our first batch of compost, mixed it in with some topsoil and spread it on the brown spots on the lawn. Fingers crossed.
I've been told that this is a 'self-healing' type of fescue and I have seen evidence of that. The thing is the ground needs to stay moist for the grass to thrive. This happens naturally (and as I found out yesterday too well when humidity is high) by letting the grass stay a bit longish, 2 1/2 to 3 inches in the hot season. But once a dead patch develops it can spread because, of course, there is nothing covering the soil any longer to keep it moist. You could cover it with mulch or soil amendment (as we did with our compost) and hope that at the very least it won't spread. I'm seeing a few little stalks of green so I remain hopeful that this grass that everyone looked at me funny and swore they didn't think it could be grown in the desert (because no one they knew had done it before) will eventually thrive.