We have the same problem with Mazie. It is funny, but she only seems to be burning out our yard - I think it may have something to do with the fertilizer my husband uses. When she periodically tinkles in the neighbors grass, it doesn't leave a mark.
My husband is the yard-man at our house, and those nasty brown spots won't do for him. We should own stock in Patchmaster

- there is always a new spot of grass growing in.
He does usually put a dose on lime on the grass in the spring to help thicken it up, I'll have to mention this thread to him for another option.
Lisa