Send them a message asking if they have writers block or are busy and you miss their work. Give them their space and say nothing Other
Add a comment on their latest chapter. Definitely no PMs, unless the author is a personal friend of yours.
I'd only ever PM if I knew them well. Normally I'd just give them space but sometimes I'll drop an encouraging comment on the last chapter they've written. Something along the lines of 'love this story, hope to see a new chapter when you're ready'. I've had stories that I've left for ages, then someone dropped a comment like that in and it inspired me to write more chapters, so it can work well as a positive motivator.
Forceably hound Dansak until he returns to the very lovely Fox and her friends! hehehe Thankfully he has captivated me with another very worthy harem of friends...
Well you're a contributor on AAFIL now, so I may hound you for more content! Seriously though, thank you for all your hounding, I really do appreciate it.
Yeah a message on the last chapter is what I'd do as well. PMs are ok but not my first thought unless we have a pre-existing relationship on the site.