its definately some form of rot. but ive seen alot of pics like this, and most reply with the same answers as its because aerogarden tends to over water at times and wasnt meant for cannabis to be grown in. However, adding hydrogen peroxide is so it can hopefully prevent the rot. What it does is its use more or less like a anti bacterial agent and helps fight all stages of rot. at what rate you say? its a drop/gallon thats for seedlings/cuttings though so you might want to put it up to maybe 2 drops and gradually add ...you need to dilute it first in water dont add it directly too the stem!
Good luck...