If you can contact your local agricultural school, they will be able to put you in contact with the best varieties for your area. What grows here would probably die instantly in your heat and temps.
My brothers are in Arizona. They have one type of grass in the winter (or is it summer?) and that goes dormant in the summer (winter?) then they either let it stay brown or they re-seed with a different grass (rye I think).
Perhaps you can plant some clover just to break up the brown areas, We have couch grass pronounced cooch!
It goes brown in winter but doesnt die off and in summer it stays green in the drought which makes up for the winter brown.
my current garden grass looks like a wig of grass sitting on the floor.
that grass was supposed to be a winter one but for some reason its dieing
and it's roots are swelling. I don't know if its bad bacteria that is doing that.
Is this your first year at the home? My lawn always goes down hill pretty bad in winter, but this is natural in a colder climate. Usually by May it is growing back with a vigor, and all greened up again.
Perhaps your lawn is just going through a natural dormancy, and you have nothing to worry about?