Screen flickers and goes black except when minimum brightness is set

I love my 24" LED Cinema Display and it has been working great until a couple of months ago. I started noticed that it didn't turn on after sleep but I finally found that it started working again if I dimmed the screen a notch. That problem sort of increased and now it's only working when it's set to the minimum brightness. If I increase the brightness it starts to flicker and then after a couple of seconds it goes black.

Everything else works as expected (USB, Camera, Charger etc).

Anyone with and idea on how I can fix this?