Has anyone made any research on this topic? Maybe the reason is lack of trust to the recently installed app or bad reputation of pushes?
You may think many users automatically prohibit the sending of push notifications and that may be the case for those who feel push notifications are intrusive and annoying. That being said, research actually indicates that app retention and engagement are up In fact, 68% of users opt-in to push notifications according to Responsys' research (http://www.responsys.com/blogs/nsm/mobile-marketing/push-notifications-marketers-new-power-tool/ ). Furthermore, research shows that engagement is up because of push notifications (http://www.emarketer.com/Article.aspx?R=1010950). . That's why marketers are focusing in on push notification strategies according to Localytics research http://www.emarketer.com/Article/Proof-App-Push-Notifications-Actually-Matter/1011149.
So if you know of an app getting poor engagement over time, the question really should be "Why is our app not engaging and what would engage users?" Also, "Is it obvious what benefit someone gets by opting into the push notifications?"