That’s a good question! I had to do quite a bit of research to find out about this one! The atmosphere of Jupiter is much different than Earth’s. It contains certain gases that are very rare in Earth’s atmosphere. Two of those gases are phosphine (PH₃, phosphorus +hydrogen) and methane (CH₄, carbon + hydrogen) (Prinn & Lewis, 1975). Phosphine and methane can be involved in a special type of chemical reaction called photodissociation (Prinn & Lewis, 1975). What this means is that the chemical molecules of these gases can be broken up when certain types of light are shined on them, specifically ultraviolet (UV) light. Basically, the light makes the phosphorus and carbon lose the hydrogen atoms that are attached to them. UV light comes from the sun, but it is not a type of light that humans can see with our eyes. When the invisible UV light breaks up the gas molecules in Jupiter’s atmosphere, the gas gives off red light that is visible to the human eye. That is why Jupiter’s “red spot” is red.

The reason that the red spot does not die out because the gas molecules that get broken up can join back together when there is not UV light being shined on them. This creates a cycle- the gas molecules get broken up and give off red light, then they join back together and can get broken up again.

reference
Prinn, R.G. & Lewis, J.S. (1975). Phosphine on Jupiter and implications for the Great Red Spot. Science, 190, 274–276.

Jupiter's great red spot is a giant storm system. I don't believe it is known why it has kept going for as long as we've seen it.

On earth, hurricanes form and then disappear in a matter of a couple of days or a week. But on Jupiter, storms can occur for years or even centuries. The Great Red Spot is a giant storm (twice as big across as planet Earth) that has lasted for around 300 years! But the storm is actually shrinking and eventually it is expected to disappear. But just as scientists started to notice that the Great Red Spot is shrinking they discovered a new red spot in addition to the Great Red Spot Jr. they already knew about. That means there are three monster storms on Jupiter!

And while it is possible that each of the storms may remain separate, they also might combine to form a super storm! We´ll just have to wait and see...