Hi @deanne and welcome to all that is THS!
No, fear not, your applicants won’t disappear. You will see this apparently random timing on messages but it isn’t a countdown to doom. I haven’t got the technical information but @MissChef is correct, it is based on GMT. When I send a message or application to an owner, mine start at “1hr ago”.
Thinking it through, that doesn’t explain why yours is counting DOWN, if it’s telling when the message was sent it should be going UP 
But, to confirm, you are not losing applicants like some sort of social media disappearing message.
Also, as you’re new, have a look at threads about responding to your applicants, such as Response to each application? to give ideas about how to manage applications and be aware that once you get five applications you can’t get any more until you decline some and “unpause” your listing.
Good luck getting the perfect sitter!