What do we know about how and why Epicurean philosophy faded in the ancient world? The general and final answer no doubt has to do with the rise of Christianity and its suppression of competitors, but it seems likely that there were other events that contributed to its decline. For example the Emperor Julian is known to have commented that the works of Epicurus were hard to find in his day, and I believe I have also read that even as early as Caesar Augustus, steps were taken against private associations that might have hindered the spread of Epicurean philosophy.
There are also discrete events in history, such as the the remark of Pompeia Plotina, wife of Trajan in about 120 AD, as to her interest and concern for the welfare of the Epicurean school.
What information do we have which a researcher looking into this topic could use as a place to start?
Note: One Place is the Cambridge Companion to Epicureanism, especially the first three essays, including this by David Sedley: