When building SLAs, you can Set Timer Indication Thresholds. It would be ideal if, those 'Timer Thresholds' would send out an email notification warning/reminder for the Assignee.
Ex. if the SLA Target is 'Resolved within 4 hours' and the Set Timer Indiction Thresholds were set as Initial = 2 hours, and Final = 1 hour...then the assignee would get an email notification with 2 hours left and then with 1 hour to go reminding them the ticket is about to breach an SLA.