Airline pricing isn't driven by econ 101 S&D. There are a couple reasons for that. The biggest reason is that it isn't perfect competitionthere are only a few airline providers and the other is that the marginal cost of flying one extra person is pretty close to $0 which is quite far from the average cost.
From a game theory perspective, and in general, one of the worst things an airline could do for itself is decrease the price of a ticket as it got closer to actual takeoff. The reason is that it would give people an incentive to wait to book.
Another thing to consider is that there are (simplifying things a bit) 2 groups of last minute ticket buyers. There are people who are value conscious and are only buying the last minute ticket because it just so happened to get very cheap. The other group is the people who found out last minute that something happened and they just have to fly. The latter group is the opposite of value conscious, they're going to buy a seat at (nearly) any price.
The first group is small because flying somewhere is not just about the cost of the plane ticket. You've got to consider the coordination of getting time off of work, you've got to have a hotel, maybe a rental car, etc. Hypothetically, if an airline offered you a seat for a flight that leaves this afternoon that goes to your favorite city, could you just drop everything else in your life to jump on that deal? Probably not, or at least that'd be the case for most people. Airlines know this and so offering last minute cheap flights isn't going to make a big difference in how many people will buy otherwise empty seats.
There's not really much to say about the people that find out last minute they just have to fly somewhere. Maybe it's a sick relative or a business event but those people can't curtail their last minute travel needs just because the price is high.
All of these reasons combine to give airlines a very strong incentive to keep prices high or the same as travel dates approach.