geez!, i'd wonder what brown sauce does to your insides as well if it can shift burnt on food
another good one is a dishwasher tablet if you've got one......cover the area with hot water pop in a tab and let it soak for a while, then scrub and repeat as necessary.
i'm currently living in as a nanny and my bosslady did this very same thing on fri night......bottom of the pan completely burnt with what should have been lovely tomato sauce( she insists on using an element to big for her pan, that wont turn down low enough.......

......i think your right about that rachel she said finally on fri night!!! doh!).
anyway, i told her to scrub as much off (she can clean up her own mess!!) cover the burnt areas with hot water and throw in a dishwash tab, let it soak, scrub and repeat as needed. i noticed yesterday morning that she had followed orders!!!! and the pan was clear and clean!.