So there are a couple different categories of abilities that I'm wondering about. They may all act the same, but I think I've noticed them behave differently in game.
DoT effects? I assume Teemo's poison will continue to tick even if Teemo dies.
Anivia's Wall? Does it last the entire duration regardless of Anivia being alive or not? Or does it disappear on death?
What about Demacian Standard cast right before death by J4? Does that continue to apply the team buff for the duration or is it gone when he dies?
Gragas' Barrels and Lux's Lucent Singularity? I've seen these disappear at death and hang around after the caster's death. My theory is that they will stick around if they have stopped moving by the time the caster has died.
Best Answer
I did some testing to show how the mechanics of abilities that create areas work. The video shows Gragas' barrel despawning if it is in motion when you die, but it stays around to do damage if it has stopped at time of death.
I also showed Anivia's skills. Crystallize (W) stays around for the full duration as expected. Flash Frost(Q) triggers automatically instead of traveling the full duration. Frostbite (E) connects regardless if you are dead or alive.
I also tested Jarvin the Fourth's Demacian Standard (E). It also stays around after death.
Finally, I tested with Lux since her Lucent Singularity was in question. Lucent Singularity (E) did land if you died while it was traveling and exploded on impact. Lucent Singularity (E) also exploded whenever you died if it was already placed. Light Binding (Q) also connected and CC's whatever is hit if you die while it is in flight. Finally, Prismatic Barrier (W) returns to your corpse even if you die while it is in flight.
We can also safely say that any ability that toggles or applies to your champion dies with your champion. Abilities that apply to other champions such as Lulu's Wild Growth, Teemo's poison, Vlad's hemoplague, Ignite, and the burn damage from Brand's passive all stay around for their duration. It is also safe to say traps and the like stay for the full duration, I'd further suggest that any time duration spell that is independent of the champion stays around (See J4's E and Anivia's W).