If you are not receiving Trainer Battle invites or you cannot see a battle icon on your friend’s profile, check the following settings:
Visit the Settings page, tap General, and tap to enable "Allow Battle Challenges with Friends". If this is unchecked, your friends will see a "Player is currently not receiving challenges" message when they try to challenge you.
Visit your device's date and time settings and set your device clock to sync automatically with local time.
If you are still having difficulty with Trainer Battles, please contact support.