Eh? Just... quantify initiative.
"Solves problems and takes initiative in his department" or whatever is a fine idea. x/5 with a little explanation. Tell him a couple times that if he abandons a task, it doesn't go away, and it doesn't become your job, and still has to get done. If you wanted to go tough-guy model, you could write him up for insubordination (abandoning his task), or if you want to go nurture mode you could tell him he has to communicate these issues with you instead of abandoning the work, and walk him through the issue but make him do it next time. You could give him a clear deadline for a task he seems to be kicking down the road, like, "need that done by 5pm today. Make it happen." And again, be ready if you need to explain unacceptable conduct, but you sound fed up with this and yet aren't establishing clear boundaries. If he wants to do the minimum, figure out your minimum, and give him that as clearly as you can. Know what your "or else" is. How much is he needed as an employee?
To build more of a carrot into it, you could give a review and ask where he sees himself in x years. Tell him what you would like to do, raises, etc., but that you need his review improved by (amount) by (performing better in the following ways) to get (better score), and tell him you're both going to brainstorm right at that moment how to get him there IF he's interested in it. Sometimes you can "unlock" the initiative in a shy or self-doubting person by showing them how empowering a given route can be. You can be instrumental in that by being particularly encouraging if he does something he previously avoided (that's a function of whether he gets pride from doing a good job). Sometimes you can unlock initiative via positive reinforcement, or even by just showing creative problem-solving strategies. You could also say candidly you know the person is (good qualities) and (capability higher than delivery), which means person is (able to be great employee) and it's because of those strengths you find (performance) dismaying, and you will need to part ways if he hasn't gotten on board by (date).
And sometimes you can't get them on board. Clarify for yourself what you're willing to accept, and then it's pretty simple to see if this guy is going to get on board, but make the effort to communicate the expectations, and you did what you can do.