Let's try to clarify that.
1. You ask the user for an e-mail address.
2. You try to send an e-mail message to that address. Perhaps you do that later or perhaps you do it before you send the response to the user -- I don't believe you explained that part, but it doesn't really matter.
3. A couple of hours later that e-mail message gets bounced back for some reason.
And now, if I understand correctly, you want to notify the user that this happened. Of course the user is no longer online and you don't have a valid e-mail address for them, so you can't contact them and tell them that.
And yes, I have a solution for that. My solution is to not care that the user gave me a bad e-mail address. Instead
you should implement a process which makes it the user's problem if they did that. I have no idea what your application is intended to do, so I can't suggest anything in detail, but the usual way to do that is to require the user to respond in some way to the e-mail you sent to "their address" before they can do anything useful on your system.