I also have the iphone 5s and am experiencing this issue. My phone is out of warranty by about 1.5 months. I went into the apple store and was told the same thing by the technician that many of you have posted. $269 for a replacement phone with a warranty of 3 months. The person who was helping me initially thought it was a software problem and specifically told me "Apple products do not have a blue screen of death. That is a PC thing." I am sorry, but as far as I can tell, my screen turns blue and the phone shuts down...that is a blue screen of death. The same thing happened to my fiance's iphone 5s and it happened with increasing frequency until it completely died. Call it what you want, but it is evidence of some type of failure.
If it is a software problem, then it's Apple's fault and they should take care of it.
If it's a hardware problem, then Apple is producing a poor quality product that costs $600 to last just over a year. They should also take care of that.
Unfortunately for now, Apple is refusing to acknowledge this issue or take responsibility. I really hope that the nightly news or viral internet chains will catch wind of this and it will become a big reputation/branding deal so that Apple can no longer make the claim that their products are "high quality and do not experience blue screens of death." I will try again at the apple store before I try to make a decision about how to move on from here, but I've never loved Apple before and had another bad experience with an ipod years ago...I am starting to fear I will have to jump ship and learn a whole new phone.
I'll update if anything changes and in the mean time if anyone finds out a fix, post it!