G'Day,
I can't speak for chrome/firefox on linux, but in Safari iOS (for the iPhone4) I've had similar issues and have had to put this aside for now in favour of just managing this via cookies rather than leaving it up to Safari iOS. Having the form tag and input type set correctly is a good start. Are your autofill options enabled? I also found that the "Go" button on the keypad doesn't seem to pick up jQuery events (at least they are not firing in my case). Another thing I came across was while I had just one form things seemed to work a bit better, but once I introduced additional forms and code, things seemed to get messed up. If you find a reliable solution, please post here (as will I).
Lastly, if you try a very basic html page without jQuery/jQuery Mobile, does it work?
cheers
Matt