There are a number of ways I can think of doing thing “in the box” i.e. without calling back to some kind of authentication server at your end.
The first one is when the application first launches it stores a record in a table that notes the date/time that the app was first launched. When the user next opens the app it checks to see if that date is < 7 days ago, if it is then the app loads if not then it says “Trial over” and exits.
This would be mega easy to “crack” as all you would need to do is either set your system time to way in the future when you first open the app or just edit the record in the table.
You could try to harden this solution by restricting access to the table or some kind of encryption of the date so its not just stored in plan text but in the end if a user if determined to crack your software then he/she will