Anything unabating is constant, ongoing, or continual, like your unabating love for your cat or your brother's unabating hunger for pizza.
To abate is to become weaker or less intense; things that are unabating don't do this, but continue on at full force. You may have unabating enthusiasm for your favorite movie franchise, first in line on opening night for each new film, but also detest the unabating heat of the endless summer days. If it never ends (or feels like it won't ever end), it's unabating.