Devices powers itself off, won't turn back on.

I bought an iPhone 5c from a guy, no water damage, no physical damage. He said he just couldn't get it to turn on one day after work. I took it home and plugged it in, after 20 minutes it booted up. If I take it off the charger it will turn itself off after about 10 minutes and the only way to turn it back on is to do a hard reset, plug it in, or disconnect and reconnect the battery. I have tried putting 3 new batteries in and the problem still exists.