Heyo,
Preflop looks good, I like the big sizing against this kind of a player. Tough spot on the flop. The stack-to-pot ratio is rather annoying at 1,1 on this board. If you happen to have any sample on villain's fold to 3bet and/or fold to 4bet, those are helpful.
If we look at this as a simplifies hand, where he pushes instead of betting small (basically I'm saying that once he bets this size, he never folds and if you call, he's going to jam turn - neither which are unrealistic assumptions although not 100%accurate), this becomes a fairly simple math and range calculation. We'll start with the math part:
- How much equity do you need in order to call the shove?
12.85 in the pot, he's jamming for 19.65-5.5=14.15
So we're calling 14.15 into a pot of 14.15 x 2 + 12.85 = 28.3 + 12.85 = 41.15
Hence, the equity needed for a break even call is: 14.15/41.15 =0.34386391 aka ~35% (rake).
Now, the tougher part begins. How much equity does AK have against his range. In game, I would start by estimating what's my equity against his value range. I'd say the main part of his value range consists of hands like JJ/QQ/TT, against which we have roughly 24%. Now the real guessing game begins, how many combos of 99/AA/KK does he have? Even more importantly, how many combos if hands that beat AK he has? AQ/AJ/A2/KJ/JT/etc, all may or may not be in his range depending how loose he is calling 4bets.
I'd recommend playing around with different ranges on equilab and taking a look on how wide the range needs to be before calling with AK is profitable. My general assumption is that it's going to be very close between calling and folding, given the info I have. I think the combination off 20% 3bet and 10AF are going to sway me towards a call this time.
Sorry for the essay.