Is There a Way to Stop Notifications from Disappearing on my Mobile Device?

Unfortunately not. This happens by design on the Mobile device if the notification is not selected for viewing from the lock screen. It's worth noting that the notifications do not disappear completely. They can still be accessed by accessing the Notifications Store. The location of the Notifications Store differs on Android and Apple devices.

How to access the Notifications store on Android Devices:

Simply swipe down the top of the screen to access notifications that have not been viewed.  They are listed under this option until viewed.

How to access the Notifications store on Apple Devices:

Similarly, swipe down the top of the screen, and then click on Notifications in the screen that drops down.  All Notifications are listed here until viewed or cleared.