Wow, verizon note 3. I have dimming issues as well. I don't care for, nor use auto brightness settings because I like a consistent light. Verizon tech didn't see any issues with my phone and suggested trying auto setting to see if dimming issue persist. I read somewhere that the phone has a safety feature and dims if phones reach too high of a temperature. I'm a fairly heavy user for periods so this could be true. A CPU Monitor app has my phone cores between 109° - 144°F, and the battery over 98°F.
from "Ms Note 3"