I've had this phone for about 2 years i think (is there a way to tell the exact days?). It was working fine until about a couple of weeks ago. It starts to randomly turned off by itself. I would call somebody or pick up a call and using text message and it would just turn off by itself. But when ac adapter is plugged in, no problem at all. Not sure if it's the battery or something else.
according to my personal experience it is because ur battery is not holding charge any more... it happens with most of the batteries..i had same problem. my battery would require 5 to 6 hours to charge and it would b empty with in 1 hour..it was in the beggining later my battery would not even open my phone without connecting it to AC adaptor.. So you can now buy a new battery.. increase the life of ur battery by doing following things:
* when u buy ur battery charge it to at least 8 to 10 hours
*do not use ur device when it is charging.
*do not use Internet sharing with ur new battery.
*do use ur old battery if u r sycn ur device with PC for long period of time.
* Use Omap clock to reduce the overclocking.. it provides bit long battery life..
hope this helps...