Why does my iPhone turn itself off

Hello,

My iPhone 5 with iOS 7.1 started turning itself off today. It will, with no warning, just shut right off, no matter whether it is plugged into the computer, or just sitting still on a desk, untouched and not running any apps.

I placed the phone into DFU mode, connected it to iTunes, and reset the phone to factory settings (this took about 15 minutes, during which time the phone remained on). However, it repeatedly shut itself off when trying to restore from a backup. I was eventually able to recover settings from iCloud, but very few of the apps have re-downloaded, as the phone is still shutting itself off again and again.

All day, the longest it has remained on without crashing has probably been less than an hour. The battery (which was new in January) has never been below 50%.

Sometimes I can turn it right back on; sometimes the power button is unresponsive until I connect it to a power source, at which point it starts right up, only to crash again minutes later.

Thanks for any leads,

Rob