Device turns off unexpectedly and does not respond

Just got my iPhone yesterday and originally it took over an hour to turn on. I was able to get it on at one point and go through the activation and syncing process. The phone turned off unexpectedly once the next morning and took over an hour to turn back on, only after plugging it into the computer and then unplugging it. Restoring it was not possible. It turned off again and I was able to boot it up withing an hour. Then now it turned off again and I haven't been able to turn it back on for over 5 hours. Help!