Mobile data connection lost on iPhone 5S
This is probably not a problem with your phone. It is up to the carrier to decide which individual handsets are actually allowed on LTE on the current cell tower. So you might just be experiencing that your carrier hasn't got enough capacity to allow everyone on LTE, so they put you on a slower network.

So even though a coverage map tells you that LTE is available - it doesn't necessarily mean it is available to everyone.